Fully discrete WENO with double entropy condition for hyperbolic conservation laws

doi:10.1080/19942060.2022.2145373

Fully discrete WENO with double entropy condition for hyperbolic conservation laws

, ; , ; , 0002-01-01 00:00:00 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 2023, VOL. 17, NO. 1, 2145373 https://doi.org/10.1080/19942060.2022.2145373 Fully discrete WENO with double entropy condition for hyperbolic conservation laws a b c Haitao Dong , Tong Zhou and Fujun Liu a b NLCFD, School of aeronautic science and engineering, Beihang University, Beijing, People’s Republic of China; School of aeronautic science and engineering, Beihang University, Beijing, People’s Republic of China; China Aerodynamic Research and Development Center, Mianyang, People’s Republic of China ABSTRACT ARTICLE HISTORY Received 15 July 2022 This paper put forward a new fully discrete scheme construction method – double entropy condition Accepted 30 October 2022 solution formula method. With that, we turn the state-of-the-art semi-discrete WENO + RK scheme into a fully discrete scheme, which is named as Full-WENO. A major difficulty of this work is that KEYWORDS we lack exact solution expressions for nonlinear equations in general cases. A feasible way we can solution formula method; go is to linearize equations and get quasi-exact solution formulas. The critical challenge is keeping fully discrete; full-WENO; both accuracy and efficiency in a scheme. Then, we get a class of new high-order schemes far better Euler equations; hyperbolic than traditional WENO schemes in the following aspects: (1) One-step to consistent high accuracy conservation laws; unsteady compressible ﬂow order in both space and time; (2) Resolution improves with the increasing CFL number; (3) Less CPU time and memory space, 1/s times of WENO with s-stage RK method in theory; (4) Excellent entropy condition satisfying property. Compared with our original work , the new method applies the more sophisticated WENO reconstruction and solves the resolution loss problems in multi-dimensional cases. The numerical tests show that the new scheme is equipped with the merits of high efficiency, high resolution and low dissipation, especially for long-time nonlinear problems. 1. Introduction Sweby, 1984), entropy condition for discontinuities (Lax, 1971), Riemann solvers (such as Roe, HLL, HLLC, etc.) Aerodynamic designs of vehicles strongly depend on (Toro, 2009), etc. With the pursuit of high-d fi elity numer- solutionsoffluidequations,however,exact solutions ical solutions to meet engineering needs, a large number can hardly be achieved due to nonlinearity of equations. of distinctive high-order schemes were constructed with At presentstage,solutions areusuallyobtainedintwo dieff rent ideas. By applying a rfi st-order TVD scheme ways: numerically through computational uid fl dynamics to equations with modified u fl xes, Harten constructed (CFD) and experimentally through wind tunnels. Wind a second-order TVD (Harten, 1983)scheme. By using tunnel experimentsare basedonrealfluids andthus limiters designed with the monotonicity preserving con- the measurement results are credible, but also costly and dition, VanLeer constructed the second-order- MUSCL dicffi ult to obtain complete information of flow. While scheme along the lines of the Godunov scheme, and now CFD can obtain more detailed flow information, there has been widely used in different elds fi (Sohn, 2005;Zhao are no guarantees that the results are reliable for all flow et al., 2019). Since the TVD condition is too strict, it is conditions. The two main factors that contribute to the not suitable for construction of higher-order schemes. credibility of CFD are algorithms and turbulence, and Some researchers also constructed higher-order schemes this paper is concerned with the former. which originated by Harten and developed by Osher From its earliest days, CFD has struggled with var- and Harten (1987)and Shu(1997) with a more relaxed ious flaws in its algorithms. One of the most serious essentially non-oscillatory (ENO) condition. Several flaws is the numerical oscillation of classical schemes years later, a compact semi-discrete weighted essentially caused by shocks. There have been ample research efforts non-oscillatory (WENO) scheme came out associated aimed at dealing with this problem, with partial success, with Runge-Kutta (RK) method which originated by LIU using techniques such as conservation forms, upwind and Osher and improved by Jiang and Shu (1996). Soon, difference, total variation diminishing (TVD) condi- it became the mainstream in high-order schemes due tion (Harten, 1983), limiters (Davis, 1987;Harten, 1983; CONTACT Fujun Liu liufujun2009@cardc.cn © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 H. DONG ET AL. to its excellent properties in shock-capturing. Under the to trace the characteristic line, which leads to a very impetus of Shu, etc., WENO schemes have been rapidly complex procedure compared with normal semi-discrete and widely used in many industries. Now, it has emerged WENO + RK. And for 1D Euler equations, their method avariety of WENO + RK schemes, such as MPWENO does not work very well due to there are three character- (Balsara & Shu, 2000), multistep WENO (Shen et al., istic lines, even computational cost is tripled (Huang & 2014;Shen&Zha, 2014), multi-resolution WENO (Wang Arbogast, 2017). et al., 2021) and in addition with Lax-Wendroff type Different from aforementioned methods, this paper WENO schemes (LW-WENO) (Li & Du, 2016;Qiu, constructs a one-step fully discrete scheme through solu- 2007;Zorío et al., 2017), ADER-WENO (Dumbser, 2014) tion formula method (Dong et al., 2002). The main point schemes united with Arbitrary high-order DErivative of solution formula method is to construct a quasi-exact Riemann (ADER) approaches (Titarev & Toro, 2002; solution formula for the partial differential equation and Toro & Hidalgo, 2009), and Semi-Lagrangian/Eulerian- then discretize it into a difference scheme. Since sys- Lagrangian (SL/EL) WENO schemes (Huang et al., 2016; tems of conservation laws are nonlinear hyperbolic type Huang & Arbogast, 2017). and their solutions may contain discontinuities, it is not Now we discuss the mentioned WENO schemes in easy to construct exact solution formulas for general ini- detail according to their time discretization method: tial values. So, first we integrate them once in space and (1) WENO + RK schemes are semi-discrete, which use obtain the Hamilton-Jacobi (HJ) equations. Although HJ multi-stage RK method to enhance time accuracy order equations are still nonlinear (in u fl x), their solutions are and avoid spurious oscillations. High-order RK meth- continuous and thus easier for discretization. Via lin- ods not only occupy a heavy burden in both computing earizing the u fl x of nonlinear HJ equations, we can obtain time and memory space, but also difficult to guarantee the quasi-exact solution formula. Applying Newton inter- TVD and robustness properties of the scheme. There- polation with limiters to the (quasi-) exact solution, Dong fore, researchers usually apply the third-order TVD-RK et al. (2002) constructed a second-order fully discrete (Jiang & Shu, 1996) method with a small CFL number. entropy condition (EC) scheme, and then also develop However, for some applications, such as numerical simu- it into a high-order version (Zhou & Dong, 2021, 2022). lation of compressible turbulence and wave propagation Since the solution formula of nonlinear equations con- problems involving long-time evolution it would be ben- taining discontinues is not easy to obtain, EC schemes eficial to use schemes which converge with higher order inlay the entropy condition of discontinuities into the flux both in time and space. (2) LW-WENO schemes are fully and thus get a quasi-exact solution after a flux lineariza- discrete, which realize time discretization by replacing all tion technique. This method can be generalized to sys- time derivatives with space derivatives through LW pro- tems, using discontinuity entropy conditions to get local cedure (Lax & Wendroff, 1960). Some can even improve quasi-exact solutions for systems (This method is easy their efficiency to twice of WENO + RK. Nevertheless, to extend to systems, simply after applying the entropy LW type schemes are extremely complex and not easy to condition of systems you can get the local quasi-exact implement in high-order situations, even need a multi- solution formula). In this paper, we leverage the solu- step strategy (Li & Du, 2016). (3) ADER-WENO schemes tion formula method and obtain a one-step fully discrete are one-step and fully discrete, which realize time dis- scheme by operating Newton interpolation with WENO cretization by LW procedure and simplified generalized construction, which is named as Full-WENO. Moreover, Riemann problem (GRP) solver. They decompose the we n fi d that it may result in some resolution loss or serve difficult problem into a sequence of m conventional Rie- oscillation in multi-dimensional problems when distin- mann problems and n fi ally achieve m-th accuracy order guishingthewaveevolutionofEulerequationsbythesin- (it can be arbitrary m-th accuracy order in theory). How- gle entropy condition adopted in EC schemes. The reason ever,thismay be alittlecostlywhenapplied to Euler is that it is only guided by velocity, however, the projected equations. Titarev and Toro (2002)toldthat, underthe velocity in multi-dimensional problems may be identified same CFL number, ADER schemes can be 2–3 times as different properties in different directions when apply- faster than WENO + RK for linear equations with con- ing Strang split technique (Strang, 1968). So, we design stantcoecffi ients, butonly50% faster for1DEuler equa- a more accurate and reliable double entropy condition, tions; (4) SL/EL-WENO schemes are fully discrete, which which applies selective ux fl reconstruction according to realize time discretization by tracing characteristic lines. both velocity and pressure. SL/EL-WENO can maintain its robustness in a relax CFL Main framework of this paper is as follows: the number, even nearly free from the limitation of CFL in second sectionintroducesgeneralconstructionsteps of cases of scalar nonlinear conservation laws (Huang et al., fully discrete schemes based on solution formula method; 2016). For scalar nonlinear cases, they apply RK method the third section proposes the concrete construction ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 3 processofFull-WENO;the fourth sectionextends new in this paper all indicates to the solution of HJ equation in scheme to three-dimensional Euler equations in curvi- Equation (2). Note: ‘ ≡ ’ means ‘define’ all over this paper. linear coordinates; the ffi th section is numerical exper- iments, includes accuracy order tests, tests of scalar 2.1. Construction of solution formula equation, one- and two-dimensional Euler equations, and sonic point test; the sixth section concludes. Dieff rentialequationisanalgebraic relation containing derivatives, while its solution is an algebraic relation con- taining no derivatives. Every part of scalar equation or 2. Construction of fully discrete schemes via matrix equations has same structure and is self-consistent solution formula method in algebraic relationships, which means one can directly This section considers the construction of schemes for construct their solution formulas. However, Euler equa- following one-dimensional Euler equations of perfect tions are vector systems, so we should use eigen-matrix gas. to transform vectors into scalars and get solution for- mulas. There are vfi e steps for constructing the solution ⎛ ⎞ ⎛ ⎞ ρ ρu formula of HJ systems: (1) scalarize (diagonalize) the sys- ⎝ ⎠ ⎝ ⎠ u + f(u) =0 ⇔ ρu + ρu + p =0, tem with eigenvector matrix; (2) linearize the ux fl via t x E u(E + p) first-order Taylor expansion or linear interpolation; (3) t x construct scalarized solution formula; (4) compose solu- ρu p p = (γ − 1)(E − ), c = γ.(1) tion formula for systems; (5) write general numerical u fl x 2 ρ by solution formula. Details of every step are as follows: Where ρ is density, u is velocity, p is pressure, E is total (1) Premultiply the HJ systems by local constant left energy, γ is specific heat ratio, c is sound speed. The eigenvectors Jacobian matrix of u fl x A = f (u)can be writteninto the form containing only velocity u and sound speed c, namely A(u,c). This matrix has complete left/right eigen- v + f(v ) = 0, t x n n vectors and eigenvalues, which can also be rewritten into v(x, t ) = v (x), the function of u and c: L(u,c), R(u,c), λ(u,c). diagonlization k k The Euler equations are actually conservation laws ↓ f(u) − −−−−−−→ ϕ (u) ≡ L f(u) ↓ (CL). First, we integrate the conservation laws once into k k L v + L f(v ) = 0, t x HJ equations. And then, we construct solution formulas (3) k n k n L v(x, t ) = L v (x). of HJ equations which will be used to obtain the numer- ical u fl x of conservative schemes. During this, there k k Where L is k-th row of matrix L, v and f are vectors, L v are two key techniques: (1) entropy condition lineariza- and L f are scalars, so Equation (3) shows the way from tion; (2) non-oscillation Newton interpolation methods. system (vector equations) to a group of scalar equations. Introducing the space integration of conserved variables, n n Note: the initial value v (x)startsfrom n-th timestep t we obtain HJ form of Euler equations. for the convenience of constructing numerical schemes. For one-dimensional Euler equations, k = 1, 2, 3, which u + f(u) = 0, t x (CL) means the k-th characteristic eld fi . n n u(x, t ) = u (x), (2) Rewrite the u fl x into linear form to linearize the udx ≡ v ↓↑ u ≡ v scalarized equations. v + f(v ) = 0, t x (HJ) (2) n n linearlization v(x, t ) = v (x). k k k k k ∗k ϕ (u) −→ λ L u−ϕ L v + L f(v ) =0, t x −→ k n k n L v(x, t ) =L v (x), Notice that the weak solution of CL systems is corre- k k k ∗k L v + λ L v − ϕ = 0, sponding to the continuous solution of HJ systems, and t x (4) k n k n L v(x, t ) = L v (x). it’s easier to get formulas of continuous solution. By using Newton interpolation, we can reconstruct solutions in k k k ∗k continuous fields from discrete data (numerical solu- Where ϕ (u) is a nonlinear function, λ L u-ϕ is a tion). This method greatly simpliefi s the derivation of linear function, so Equation (3) shows the way from non- schemes, which is easier for readers to understand the linear equation to linear equation. Note: L u is a scalar k ∗k essence of the algorithm. By the way, the solution formula variable, λ and ϕ are constants. 4 H. DONG ET AL. (3) The solution formula for the linearized scalarized yet. L u ¯ in above equations can be simplified into j+ equations can be given scalar form due to L are local constants. And for each k k k ∗k diagonalized solution formula L v + λ L v − ϕ =0, k theequations areall same,the k can be omitted. t x −→ k n k n L v(x, t ) =L v (x). n n υ − υ (x − λτ ) j+ j+ k n+1 k n k ∗k 2 k 2 × L v (x) = L v (x − λ τ) + τϕ .(5) L u ¯ = u ¯ ≡ j+ λτ j+ k k ∗k If we denote υ ≡ L v, a ≡ λ , f ∗≡ ϕ , the equation λτ of Equation (5) can be written as υ + aυ – f = 0, ↓ ν ≡ ↓ t x this linear equation has an exact solution expression as n n υ − υ n + 1 n 1 1 υ(x,t ) = υ (x-aτ)+τf ∗,justthe same as thesolu- j+ j+ −ν 2 2 n + 1 n u ¯ ≡ .(8) tion expression of Equation (5), where τ ≡ t -t is j+ νh time step size. Where the values with subscript j+1/2+ν are gener- ally not at the grid nodes, so we need an interpolation (4) The solution formula for systems can be composed method to construct the initial value υ (x). These are the by right eigenvectors whole idea of solution formula and general expression of n+1 1 1 n 1 ∗1 v (x) = R (L v (x − λ τ) + τϕ ) numerical u fl x. Then we just need to solve two key tech- 2 2 n 2 ∗2 niques: (1) Discontinuity entropy condition linearization + R (L v (x − λ τ) + τϕ ) techniquetodecidelocal constantsinsolutionformu- 3 3 n 3 ∗3 + R (L v (x − λ τ) + τϕ ).(6) las; (2) Non-oscillation Newton interpolation method to discrete numerical solution and to reconstruct the ini- Thesefoursteps abovecan reachthe solution formula tial value υ (x), and this paper we apply the thought of k k k ∗k expression of HJ systems. Where L , R , λ , ϕ are all WENO reconstruction. undetermined local constants, which will be given by dis- continuity entropy condition in next section. Note: R is 2.2. Double entropy condition linearization method the k-th column of matrix R. There exist global linearization methods for scalar non- (5) From solution formula Equation (6), we can obtain linear function, and thus exact solution formula can be the general expression of numerical u fl x for conser- constructed. However, the global linearization for non- vative scheme linear vector function cannot be achieved easily, so we try n n to construct quasi-exact formula by local linearization. v −v 1 1 j+ j− n 2 2 Local linearization is adopted at the interface of grid cells u ≡ n+1 n τ n n ˜ ˜ u = u − (f − f ) ←− [u , u ], which is a discontinuity formed by bilateral 1 1 j j+1 j j j+ j− 2 2 values. By the way, the linearization method for nonlin- n+1 n v = v − τf 1 ear vector is not unique, now we apply following double 1 1 j+ j+ j+ 2 2 entropy condition to determine the local constants (coef- n n +1 v −v ficients and constants of linear functions) in Equation 1 1 j+ j+ 2 2 f = , 1 (7): j+ u-entropy condition:Weapply velocity entropycondi- tion to obtain the local constants L, R, λ due to they n +1 k k n k ∗k v = R (L v (x − λ τ) + τϕ ), arematrixasawholethatdonot splitwithcharacter- j+ j+ k k k k=1 istic fields. Specifically, the local constants L , R , λ are constructed with the guide of bilateral velocity relation- ship u ≷u at arbitrary discontinuities [u , u ], so that L R L R we can scalarize the system. k k k ∗k f = R (λ L u ¯ − ϕ ), 1 1 j+ j+ √ √ 2 2 k=1 ρ u + ρ u L L R R ⎨ √ √ u > u L R ρ + ρ L R n n k v −v (x −λ τ) u ≡ , j+ j+ ⎩ ρ u +ρ u L L R R k k 2 u ≤ u L u ¯ ≡ L .(7) L R 1 ρ +ρ L R λ τ j+ 2 ⎧ √ √ ρ H + ρ H L L R R √ √ u > u L R ρ + ρ L R Note: Equation (7) is just the quasi-exact solution in a H ≡ , u +u L R H( ) u ≤ u numerical scheme form, it is not the n fi al discrete scheme L R 2 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 5 and pressure in union will be more accurate, especially for A ≡ A(u, c), multi-dimensional problems. The reason is the projected 2 L ≡ L(u, c), c = (γ − 1)(H − ) → (9) velocity may be identified as different properties (com- R ≡ R(u, c), pression wave or rarefaction wave) in different directions {λ }≡ λ(u, c). during applying Strang split technique (Strang, 1968), Concretely, we apply Roe means for u > u ,whilethe L R especially at the slip lines (contact discontinuities in 1D characteristic linestendtoconvergeintoashock. Andwe problems). Concretely, the scheme may apply Roe mean apply arithmetic means for u ≤ u , while the character- L R for compression waves in one direction and apply recon- n+1 isticlinetendstodivergeintorarefactionwaves.Withthis structed u in the other direction for rarefaction waves, j+ procedure, we can diagonalize the system. And here λ which may lead to a resolution loss or severe oscillation, will be used in following steps. such as that shown in section 5.4, Double Mach reflection p-entropy condition: We apply pressure entropy con- ∗k problem. With double entropy condition, we can more dition to obtain the local constants ϕ due to they can be precisely distinguish compression waves and rarefaction customized according to whether the characteristic elds fi are rarefaction waves or compression waves. Specicfi ally, waves. Moreover, we customize the reconstruction pro- ∗k the local constants ϕ are constructed with the guide posal for three efi lds of Euler equations. Due to the of pressure relationship p ≷ p at arbitrary discontinu- L/R second field of 1D Euler equations (second, fourth and ities [u , u ], so that we can linearize the scalarized u fl x. L R fifth fields in 3D Euler equations) is equipped with lin- Where p is an approximate middle pressure that can be ear features, we safely apply the high-order reconstructed obtained by discontinuity decomposition from bilateral n+1 u . status. j+ 2γ ⎛ ⎛ ⎞ ⎞ γ −1 u −u c c L R L R + + ⎜ ⎜ ⎟ ⎟ 2 γ −1 γ −1 p = max 0, , ⎝ ⎝ ⎠ ⎠ γ −1 γ −1 c 1 c 1 L 2γ R 2γ γ −1 p γ −1 p L R 3. Construction of full-WENO via solution u +u f(u )+f(u ) 1 L R L R formula method λ − , p > p , 2 2 ∗1 1 ϕ ≡ L , u +u u +u 1 L R L R λ − f( ), p ≤ p , 3.1. Initial value reconstruction 2 2 ∗2 2 2 u +u u +u L R L R ϕ ≡ L λ − f( ) , Using a Newton interpolation with WENO reconstruc- 2 2 u +u f(u )+f(u ) tion (Jiang & Shu, 1996), we construct the initial value 3 L R L R λ − , p > p , 2 2 ∗3 3 k n k n 1 ϕ ≡ L . (10) function L v (x)frominitial data L v , i = j + .And u +u u +u 3 L R L R λ − f( ), p ≤ p , n k n 2 2 for writing convenience, we denote υ (x) ≡ L v (x). (2r−1)-th Newton interpolation will occupy 2r−1grids, Concretely, we apply Roe means for p > p ,since these L/R consider the upwind condition and two numerical ux fl es, compression waves may evolve into shocks. Otherwise, the scheme will occupy 2r + 1 grids in total. According to we apply arithmetic means. This step is to linearize the WENO reconstruction, we weight several r-th stencils by scalarized equations. Note: Equations (7) and (8) and n n performing with smoothness indicators, which not only Equations (9) and (10) with [u , u ] ≡ [u , u ]make L R j j+1 helps to reach (2r−1)-th order at smooth regions but also up the so-called Solution Formula Method of this paper. can avoid the oscillation caused by high-order interpo- Above techniques which determine the local constants lation at risk regions. For example, a fifth-order stencil by predicting the discontinuities evolve into shocks or can be weighted by three third-order stencils, a seventh- rarefaction waves are named as entropy condition. Sim- order stencil can be weighted by four fourth-order sten- ilar to the statement in (Zhou & Dong, 2021, 2022), cils. Totally, this process will provide basic (2r−1)-th Equations (9) and (10) that provides basic second order orderinspace forFull-WENO scheme.Wegivethe spe- in temporal discretization for schemes in solution for- cicfi expressionsnow.(Note:wegivefifthand seventh mula method can be named as baseline double entropy schemes as examples here because it’s not easy to give the condition linearization. If we replace the arithmetic simple general expressions) means (u + u )/2 into a reconstructed high-order ver- L R n+1 sion u , it can provide higher-order nonlinear tempo- j+ ral accuracy order for scheme, details are shown in next section. 3.1.1. Initial value reconstruction for fifth order By the way, compared to the single entropy condition full-WENO (Full-WENO5) guided only by velocity applied in (Zhou & Dong, 2021, The fully discrete interpolation and weighted coefficients 2022), thedoubleentropy conditionguidedbyvelocity of Full-WENO5 can be given as. 6 H. DONG ET AL. 2 3 4 3−ν −1−ν 2−ν 1−ν ⎪ u + + + + , a ≥ 0, j 1 j 1 j weighted by 3 third order stencils ⎨ 5 4 3 2 j− j− 2 −→ u ¯ = j+ ⎪ ←− 2 3 4 −3−ν 1−ν −2−ν −1−ν u + + + + , a ≤ 0, finial fifth order stencils ⎩ j+1 j+1 3 j+1 5 4 3 2 j+ j+ (1)− (2)− (3)− − − − γ u + γ u + γ u ,a ≥ 0, 1 1 2 1 2 1 j+ j+ j+ 2 2 2 u ¯ = j+ (1)+ (2)+ (3)+ + + + 2 ⎩ γ u + γ u + γ u ,a ≤ 0. 1 1 2 1 2 1 j+ j+ j+ 2 2 2 a≥0 ⎪ (1)− 2−ν 1−ν u = u + + , ⎪ 1 ⎪ 1 j−1 3 2 j− ⎪ ⎧ j+ ⎪ − 1 γ = (1 + v)(2 + v), ⎪ ⎪ 1 20 ⎨ ⎨ (2)− 2 2−ν 1−ν − u = u + + , j 1 γ = (3 − v)(2 + v), 1 j 3 2 2 j− ⎪ j+ ⎪ ⎪ ⎩ ⎪ − γ = (3 − v)(2 − v), (3)− ⎪ 2 −1−ν 1−ν u = u + + , ⎩ j 1 1 j+1 3 2 j+ j+ a≤0 (1)+ 2 1−ν −1−ν u = u + + , ⎪ 1 j+1 ⎪ 1 j 3 2 j+ ⎪ ⎧ j+ ⎪ 2 + 1 γ = (2 + v)(3 + v), ⎪ ⎪ 1 20 ⎨ ⎨ (2)+ −2−ν −1−ν 2 + 1 u = u + + , j+1 3 γ = (2 − v)(3 + v), (11) 1 j+1 3 2 2 j+ ⎪ j+ ⎪ ⎪ 2 ⎩ ⎪ + γ = (2 − v)(1 − v). (3)+ ⎪ 2 −2−ν −1−ν u = u + + , ⎩ j+1 3 1 j+2 3 2 j+ j+ k m Final interpolation of Full-WENO5 can be written as Where ν = aτ/h, a = λ , is the m-th order differ- n n 2 3 ence, = u − u , = − , = 1 1 1 j+1 j j 1 j+ j+ j− j+ 2 2 2 2 − 2 2 ⎪ 1 − , etc. For nonlinear cases, we need to con- ⎪ 2 j +1 j − (ε+β ) (1)− sider the accuracy of λ,and we achieveitthrough theflux − − − ⎪ 1 ⎪ γ γ γ 1 2 3 j+ + + 2 reconstruction technique in section 3.2. The expression 2 2 2 ⎪ − − − ⎪ (ε+β ) (ε+β ) (ε+β ) ⎪ 1 2 3 of u ¯ in Equation (11) is deduced via Newton interpo- 1 ⎪ ⎪ − j+ ⎪ γ ⎪ 2 lation of υ (x), see Appendix A for detailed process. ⎪ (ε+β ) (2)− ⎪ + u − − − ⎪ 1 Smoothness indicatorfor Full-WENO5,from(Jiang& γ γ γ 1 2 3 j+ + + 2 ⎪ 2 2 2 − − − Shu, 1996) (ε+β ) (ε+β ) (ε+β ) 1 2 3 a≥0 3 ⎪ 2 (ε+β ) ⎪ (3)− − ⎪ 13 2 + u ,a ≥ 0, ⎪ − − − ⎪ β = (u − 2u + u ) 1 j−2 j−1 j ⎪ γ γ γ ⎪ 1 12 j+ ⎪ 1 2 3 ⎪ + + 2 2 2 2 ⎪ 1 2 − − − (ε+β ) (ε+β ) (ε+β ) ⎪ + (u − 4u + 3u ) , j−2 j−1 j 1 2 3 u ¯ = j+ ⎨ − 13 γ 2 2 ⎪ 1 β = (u − 2u + u ) j−1 j j+1 12 ⎪ 2 ⎪ (ε+β ) (1)+ ⎪ 1 ⎪ u + + + + (u − u ) , ⎪ 1 ⎪ j−1 j+1 4 ⎪ γ γ 1 2 3 j+ ⎪ + + 2 ⎪ 2 2 2 + + + ⎪ − 13 2 (ε+β ) (ε+β ) (ε+β ) 1 2 3 β = (u − 2u + u ) ⎪ ⎪ j j+1 j+2 3 12 ⎪ ⎪ + 1 2 γ + (3u − 4u + u ) , ⎪ j j+1 j+2 4 2 ⎪ (ε+β ) (2)+ ⎪ + u a≤0 + + + γ γ γ ⎪ 1 2 3 j+ + + 2 ⎪ 2 2 2 + + + 13 2 ⎪ (ε+β ) (ε+β ) (ε+β ) β = (u − 2u + u ) 1 2 3 j−1 j j+1 1 ⎪ ⎪ 12 1 2 ⎪ + (u − 4u + 3u ) , 3 j−1 j j+1 ⎪ ⎪ 4 ⎪ 2 ⎨ + (ε+β ) (3)+ + 13 1 ⎪ 2 2 + u ,a ≤ 0. ⎪ + + + β = (u − 2u + u ) + (u − u ) , 1 j j+1 j+2 j j+2 ⎪ γ γ γ 12 4 j+ ⎪ 1 2 3 ⎩ + + 2 ⎪ 2 2 2 + + + + 13 (ε+β ) (ε+β ) (ε+β ) ⎪ 1 2 3 ⎪ β = (u − 2u + u ) j+1 j+2 j+3 ⎪ 3 12 (13) + (3u − 4u + u ) . j+1 j+2 j+3 (12) ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 7 Where is a positive real number introduced to avoid the 3.1.2. Initial value reconstruction for seventh order −6 denominator becoming zero, and we will take = 10 full-WENO (Full-WENO7) in later tests. Equations (12) and (13) come from refer- The fully discrete interpolation and weighted coefficients ence (Jiang & Shu, 1996). of Full-WENO7 can be given as. 2 3 4 5 6 4−ν −2−ν 3−ν −1−ν 2−ν 1−ν u + ( + ( + ( + ( + ( + ) ) ) ) ) , a ≥ 0, ⎪ j j 1 j 1 j 7 6 5 4 3 2 j− j− j− 2 2 u ¯ = j+ ⎪ 2 3 4 5 6 −4−ν 2−ν −3−ν 1−ν −2−ν −1−ν u + ( + ( + ( + ( + ( + ) ) ) ) ) , a ≤ 0, ⎩ j+1 j+1 3 j+1 3 j+1 7 6 5 4 3 2 j+ j+ j+ 2 2 weighted by 4 fourth order stencils −→ ←− finial seventh order stencils ⎪ − (k)− γ u ,a ≥ 0, j+ k=1 u ¯ = j+ (k)+ 2 + γ u ,a ≤ 0. k 1 j+ k=1 a≥0 (1)− 3−ν 2−ν 1−ν n 2 3 u = u + ( + ( + ) ) , ⎪ 1 1 j j−1 3 4 3 2 j− ⎪ j+ j− ⎪ 2 2 (2)− ⎪ −1−ν 2−ν 1−ν n 2 3 u = u + ( + ( + ) ) , ⎨ 1 1 j j 1 4 3 2 j− j+ j− 2 2 (3)− 3 −1−ν 2−ν 1−ν n 2 u = u + ( + ( + ) ) , ⎪ 1 1 j j 1 4 3 2 j− ⎪ j+ j+ ⎪ 2 2 (4)− ⎪ n 3 −2−ν −1−ν 1−ν u = u + ( + ( + ) ) , ⎩ 1 1 j j+1 3 4 3 2 j+ j+ j+ 2 2 − 1 ⎪ γ = (1 + v)(2 + v)(3 + v), 1 210 − 1 γ = (4 − v)(2 + v)(3 + v), ⎪ − 1 γ = (4 − v)(3 − v)(3 + v), 3 70 − 1 γ = (4 − v)(3 − v)(2 − v), a≤0 (1)+ n 2 3 2−ν 1−ν −1−ν u = u + ( + ( + ) ) , ⎪ 1 1 j+1 j 1 ⎪ 4 3 2 j+ ⎪ j+ j− 2 2 ⎪ (2)+ n 2 3 1−ν −2−ν −1−ν u = u + ( + ( + ) ) , ⎪ 3 1 j+1 j+1 1 ⎨ 4 3 2 j+ j+ j+ 2 2 ⎪ (3)+ n 2 3 1−ν −2−ν −1−ν u = u + ( + ( + ) ) , 1 j+1 j+1 3 4 3 2 ⎪ j+ j+ j+ ⎪ 2 2 (14) (4)+ −3−ν −2−ν −1−ν ⎪ n 2 3 u = u + ( + ( + ) ) , 1 j+1 j+2 5 4 3 2 j+ j+ j+ ⎧ 2 2 γ = (2 + v)(3 + v)(4 + v), 1 210 ⎨ + 1 γ = (3 − v)(3 + v)(4 + v), γ = (3 − v)(2 − v)(4 + v), ⎩ + 1 γ = (3 − v)(2 − v)(1 − v). 4 210 The expression of u ¯ in Equation (14) is deduced via j+ Newton interpolation of υ (x), the detailed process is the same as fifth order case and is omitted here. 8 H. DONG ET AL. Smoothness indicator for Full-WENO7, from (Balsara designed scheme is fully high order for linear equations. &Shu, 2000) Now we show the fully high-order version in nonlinear cases. If we want high-order nonlinear accuracy, we need a≥0 n + 1 to linearize the u fl x with exact solution at time t for β = u (547u − 3882u − 4642u ⎪ 1 j−3 j−3 j−2 j−1 rarefaction waves and compression waves before shocks −1854u ) + u (7043u − 1724u + 7042u ) ⎪ j j−2 j−2 j−1 j formed.However,wecannotgetthenonlinearexactsolu- +u (11003u − 9402u ) + 2107u , j−1 j−1 j j tion, so we need to reconstruct the quasi-exact solution n+1 n ⎪ β = u (267u − 1642u + 1602u − 494u ) 2 j−2 j−2 j−1 j j+1 u by thedataattime t to replace the arithmetic ⎪ j+ +u (2843u − 5966u + 1922u ) 2 j−1 j−1 j j+1 ⎨ 2 means in Equations (9) and (10). More specifically, the +u (3443u − 2522u ) + 547u , j j j+1 j+1 n+1 quasi-exact solution u origins from solution formula β = u (547u − 2522u + 1922u − 494u ) 3 j−1 j−1 j j+1 j+2 j+ ⎪ +u (3443u − 5966u + 1602u ) method for conserved variables, which actually is the j j j+1 j+2 ⎪ n +1 ⎪ +u (2843u − 1642u ) + 267u , first-order derivative of v in Equation (7): j+1 j+1 j+2 ⎪ j+2 j+ β = u (2107u − 9402u + 7042u − 1854u ) ⎪ 4 j j j+1 j+2 j+3 +u (11003u − 17246u + 4642u ) j+1 j+1 j+2 j+3 n+1 1 1 n 1 2 2 n 2 u = R (L v (x − λ τ)) + R (L v (x − λ τ)) +u (7043u − 3882u ) + 547u , 1 1 1 x x j+2 j+2 j+3 j+3 j+ j+ j+ 2 2 a≤0 3 3 n 3 + R (L v (x − λ τ)). (17) j+ β = u (2107u − 9402u + 7042u ⎪ 1 j+1 j+1 j j−1 −1854u ) + u (11003u − 17246u ⎪ j−2 j j j−1 k k k n+1 ⎪ Here the initial L , R , λ in u can be acquired by +4642u ) + u (7043u − 3882u ) ⎪ j−2 j−1 j−1 j−2 j+ 2 2 +547u , j−2 baseline u-entropy conditioninEquation(9) ⎪ β = u (547u − 2522u + 1922u k k k 2 j+2 j+2 j+1 j Then the reconstructed L , R , λ can be given by −494u )+u (3443u −5966u + 1602u ) ∗k j−1 j+1 j+1 j j−1 u-entropy condition and ϕ can be given p entropy ⎨ 2 +u (2843u − 1642u ) + 267u , j j j−1 condition. j−1 β = u (267u − 1642u + 1602u 3 j+3 j+3 j+2 j+1 ⎪ n n n n ⎪ −494u ) + u (2843u − 5966u + 1922u ) ρ u + ρ u j j+2 j+2 j+1 j j j j+1 j+1 ⎪ ⎪ n n ⎪ ⎪ u > u ⎪ ⎨ j j+1 n n ⎪ +u (3443u − 2522u ) + 547u , j+1 j+1 j ρ + ρ ⎪ j j j+1 u ≡ , β = u (547u − 3882u − 4642u ⎪ ⎪ 4 j+4 j+4 j+3 j+2 ⎪ n+1 n n ⎪ ⎩ u u ≤ u 1 j j+1 −1854u ) + u (7043u ⎪ j+ j+1 j+3 j+3 ⎪ ⎧ −1724u + 7042u ) j+2 j+1 n n n n ρ H + ρ H j j j+1 j+1 2 ⎪ n n +u (11003u − 9402u ) + 2107u . u > u j+2 j+2 j+1 ⎪ j+1 n n j j+1 ρ + ρ j j+1 (15) H ≡ , n+1 n n H u u ≤ u 1 j j+1 Final interpolation of Full-WENO7 can be written as j+ A ≡ A(u, c), ⎪ k ⎪ 4 − 2 L ≡ L(u, c), ⎪ (ε+β ) (k)− ⎪ k c = (γ − 1)(H − ) → u ,a ≥ 0, ⎪ − 2 4 ⎪ R ≡ R(u, c), ⎪ γ j+ ⎪ ⎪ k k=1 ⎩ ⎨ − λ ≡ λ(u, c), (ε+β ) k=1 u ¯ = (16) 2γ + ⎛ ⎛ ⎞ ⎞ j+ ⎪ γ ⎪ k γ −1 n n n n u −u c c ⎪ 2 4 + j j+1 j j+1 (ε+β ) ⎜ ⎜ + + ⎟ ⎟ ⎪ (k)+ 2 γ −1 γ −1 u ,a ≤ 0. ⎜ ⎜ ⎟ ⎟ ⎪ + p = max 0, , ⎪ γ γ −1 γ −1 ⎝ ⎝ n n ⎠ ⎠ j+ ⎪ k c c k=1 ⎩ 2 j j+1 1 1 2γ 2γ ( ) + ( ) n n (ε+β ) k=1 γ −1 p γ −1 p j j+1 ⎛ ⎞ n n n n u +u f(u )+f(u ) j j+1 j j+1 Equations (15) and (16)come from reference (Balsara & 1 n λ − p > p 2 2 j ⎜ ⎟ Shu, 2000). ∗1 1 ⎜ ⎟ ϕ ≡ L , ⎝ ⎠ n+1 n+1 1 n λ u − f u p ≤ p 1 1 j j+ j+ 2 2 3.2. Flux reconstruction for full-WENO ∗2 2 2 n+1 n+1 ϕ ≡ L λ u − f u , 1 1 This processistoreach designed accuracy fornonlin- j+ j+ 2 2 ⎛ ⎞ n n n n ear equations, to be exact, we need to obtain the local u +u f(u )+f(u ) j j+1 j j+1 3 n λ − p > p k k k ∗k j+1 2 2 constants L , R , λ , ϕ for the composition of nal fi ⎜ ⎟ ∗3 3 ⎜ ⎟ ϕ ≡ L . (18) ⎝ ⎠ numericalflux.Wehavegiven thebaselineentropy con- n+1 n+1 3 n λ u − f u p ≤ p 1 1 j+1 dition linearization in section 2.2 and guarantee that the j+ j+ 2 2 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 9 k k k Due to the accuracy order of L , R , λ getting from in section 3.1 to maintain the non-oscillation property. baseline u-entropy conditionisnot equaltothe recon- Details are shown as follows. n+1 structed u ,itwillinufl encethe accuracy to some j+ 3.2.1. Flux reconstruction for r = 3, full-WENO5 content. So, theoretically, iteration is needed for getting a n+1 2 2 2 2 ∗ ∗ more accurate u .However,wedonotrecommendthis if (β = min(β , β , β )) ⇒ u = u , j+ k 1 2 3 k 2 j+ ⎧ 2 ⎡ ⎤ process owing to the complexity of Euler equations and −(−v)(1−v) u + j 1 dv 2! ⎪ j− consideration of computing burden. Actually, the numer- ⎪ ⎢ ⎥ ⎪ ⎢ ⎥ −(−v)(1−v)(2−v) ⎪ 2 d ical experiments tell that the error gently corrected by ⎪ ⎢ ⎥ j−1 dv 3! ⎪ ⎢ ⎥ iteration contributesfew inufl encesinthe vast major- ⎪ ⎢ ⎥ d −(−v)(1−v) ⎪ ⎢ ⎥ u + j 1 dv 2! ity of tests, except for testing nonlinear accuracy order. ⎪ ⎢ j− ⎥ ⎪ ⎢ ⎥ , a ≥ 0, −(−v)(1−v)(2−v) Moreover, the greatest influence factors on resolutions ⎪ ⎢ 2 d ⎥ ⎪ ⎢ ⎥ dv 3! k ∗k ⎪ ⎪ ⎢ ⎥ are the reconstructed λ ϕ , so we even do not need to d −(−v)(1−v) ⎪ ⎢ ⎥ k k ⎪ u + j 1 ⎪ ⎢ ⎥ reconstruct L , R for almost all tests. It means we can dv 2! ⎪ j+ ⎪ 2 ⎡ ⎤ ⎣ ⎦ k k directly use the L , R in Equation (17) which is calcu- u d −(−1−v)(−v)(1−v) ⎨ 2 j+1 dv 3! ⎣ ⎦ ⎡ ⎤ lated by baseline entropy condition. By the way, we also u ≡ 2 d −(−1−v)(−v) u + ∗ j+1 1 dv 2! keep in line with the above statement in following numer- u ⎪ j+ 3 ⎪ ⎢ ⎥ ⎪ ⎢ ⎥ ical experiments. (Note: we can get the entropy condition ⎪ −(−1−v)(−v)(1−v) 2 d ⎪ ⎢ ⎥ dv 3! ⎪ ⎢ ⎥ linearization formula for the scalar equation if we remove ⎪ ⎪ ⎢ ⎥ ⎪ d −(−1−v)(−v) ⎪ ⎢ ⎥ u + j+1 the eigenvectors LR and use the eigenvalue for determin- ⎪ dv 2! ⎪ ⎢ j+ ⎥ ⎪ 2 ⎪ ⎢ ⎥ , a ≤ 0. ing the selection between Roe means and reconstruction, ⎪ ⎢ d −(−2−v)(−1−v)(−v)⎥ ⎪ ⎢ ⎥ j+1 dv 3! see details in Zhou & Dong, 2021, 2022) ⎪ ⎢ ⎥ −(−1−v)(−v) n+1 ⎪ ⎢ d ⎥ k ∗ u + Denote L u ≡ u ,accordingtothe accuracy 3 j+1 ⎪ ⎢ ⎥ 1 1 dv 2! j+ j+ j+ ⎪ ⎣ 2 ⎦ 2 2 ⎪ ⎩ d −(−2−v)(−1−v)(−v) order analysis in (Zhou & Dong, 2021, 2022), final accu- j+2 dv 3! racy order of scheme via solution formula method = min (19) (accuracy order of initial value reconstruction, 2 × accuracy order of ux fl reconstruction). So, we can obtain The expression of u in Equation (19) is deduced j+ designed order by r-th interpolation of u∗ + .While j 1 via Newton interpolation of υ (x), see Appendix A for doing r-th Newton interpolation for initial value υ (x)in detailed process. The expansion form of Equation (19) is section 3.1, we can obtain their corresponding rfi st-order in Appendix A derivative function u (x). Inserting x = x -νh,weget j+ 3.2.2. Flux reconstruction for r = 4, full-WENO7 the n fi al formula. And we use the smoothness indicators 2 2 2 2 2 ∗ ∗ if (β = min(β , β , β , β )) ⇒ u = u , k 1 2 3 4 k j+ ⎡ ⎤ d −(−v)(1−v) d −(−v)(1−v)(2−v) d −(−v)(1−v)(2−v)(3−v) 2 3 u + + + j 1 dv 2! j−1 dv 3! 3 dv 4! ⎪ j− j− ⎪ 2 ⎢ ⎥ ⎢ −(−v)(1−v) −(−v)(1−v)(2−v) −(−1−v)(−v)(1−v)(2−v) ⎥ d 2 d 3 d ⎪ u + + + ⎢ j ⎥ ⎪ j 1 dv 2! dv 3! dv 4! j− j− ⎢ ⎥ ⎢ ⎥ , a ≥ 0, d −(−v)(1−v) 2 d −(−v)(1−v)(2−v) 3 d −(−1−v)(−v)(1−v)(2−v) ⎢ ⎥ u + + + ⎪ 1 j 1 ⎪ dv 2! dv 3! dv 4! ⎢ j− ⎥ ⎡ ⎤ j+ ⎪ 2 ⎣ ⎦ ⎪ d −(−v)(1−v) d −(−1−v)(−v)(1−v) d −(−2−v)(−1−v)(−v)(1−v) 2 3 u + + + ⎨ j 1 ∗ dv 2! j+1 dv 3! 3 dv 4! ⎢ ⎥ j+ u j+ ⎢ ⎥ 2 ≡ ⎡ ⎤ ∗ −(−1−v)(−v) −(−1−v)(−v)(1−v) −(−1−v)(−v)(1−v)(2−v) ⎣ ⎦ d 2 d 3 d u ⎪ u + + + j+1 1 3 ⎪ j 1 dv 2! dv 3! dv 4! ⎪ j+ j− ⎪ 2 ⎢ ⎥ u ⎪ 4 ⎪ −(−1−v)(−v) −(−2−v)(−1−v)(−v) −(−2−v)(−1−v)(−v)(1−v) ⎢ d 2 d 3 d ⎥ ⎪ u + + + ⎢ j+1 ⎥ ⎪ j+1 1 dv 2! dv 3! dv 4! ⎪ j+ ⎢ j+ ⎥ ⎢ ⎥ , a ≤ 0. d −(−1−v)(−v) d −(−2−v)(−1−v)(−v) d −(−2−v)(−1−v)(−v)(1−v) 2 3 ⎢ ⎥ ⎪ u + + + j+1 3 ⎪ dv 2! j+1 dv 3! 3 dv 4! ⎢ j+ ⎥ ⎪ j+ ⎣ ⎦ ⎪ −(−1−v)(−v) −(−2−v)(−1−v)(−v) −(−3−v)(−2−v)(−1−v)(−v) d 2 d 3 d u + + + ⎩ j+1 3 j+2 5 dv 2! dv 3! dv 4! j+ j+ (20) 10 H. DONG ET AL. Where the smoothness indicators β in u fl x applied in initial value reconstruction for saving com- reconstruction can also be calculated by Equation (12) puting time, which means the β applied in initial value (for Full-WENO5) and Equation (15) (for Full-WENO7). reconstruction and ux fl reconstruction are same. The With the minimum squared value of β ,wecan getthe expansion form of Equation (20) is in Appendix A. smoothest stencil for ux fl reconstruction and avoid the Additionally, for some cases with high demand in suspicious oscillations at risk regions as much as possi- robustness, such as Test 12 Double Mach reflection in ble. Moreover, the β we get in this step can be directly section 5.4, u fl x reconstruction requires an order reduc- tion.We’dliketodoasfollows. 2 2 2 2 u , if < 0 ∩ < 0 ∩ < 0 , a ≥ 0, ⎪ 1 1 j−1 j j j+1 j− j+ 2 2 u = (21) j+ ⎪ 2 ⎪ 2 2 2 2 u , if < 0 ∩ < 0 ∩ < 0 , a ≤ 0. j+1 1 3 j j+1 j+1 j+2 j+ j+ 2 2 Equation (21) means we do the order reduction (Strang, 1968) and turn it into three one-dimensional whilemiddleconvexity is dieff rent with twoendsof equations. it (nonconvex-convex-nonconvex, convex-nonconvex- ⎛ ⎞ convex). This method balances robustness and resolution ⎜ ⎟ ρu to some extent. ⎜ ⎟ ⎜ ⎟ u + f (u) = 0 ⇔ ρv Note 1: The entropy condition of solution formula e ⎜ ⎟ ⎝ ⎠ e=ξ,η,ζ ρw method and smoothness indicator of WENO guarantee high resolution and non-oscillation properties together. ⎛ ⎞ Then Full-WENO can be composed as follows. ρ(e u + e v + e w) x y z ⎜ ⎟ Full-WENO5 is composed of Equations (7–13) and (18 ρ(e u + e v + e w)u + e p(u) x y z x ⎜ ⎟ ⎜ ⎟ and 19). + ρ(e u + e v + e w)v + e p(u) = 0, x y z y ⎜ ⎟ ⎝ ⎠ e=ξ,η,ζ ρ(e u + e v + e w)w + e p(u) Full-WENO7 is composed of Equations (7–10) and x y z z (14–16) and (18 and 20). (e u + e v + e w)(E + p(u)) x y z Note 2: If we remove the u fl x reconstruction, take ν = 0 ⇓ dimensional splitting and associate it with RK method, the scheme in this paper ⎛ ⎞ will degrade to a normal semi-discrete WENO + RK ⎜ ⎟ ρu ⎜ ⎟ scheme. ⎜ ⎟ u + f (u) = 0 ⇔ ρv t e ⎜ ⎟ Note 3: If we remove the eigenvectors, solution for- ⎝ ⎠ ρw mula, and entropy condition, then use RK method to tracethe characteristic line,wecan getthe SL/EL-WENO ⎛ ⎞ (Huang et al., 2016) for scalar nonlinear equation. What ρ(e u + e v + e w) x y z ⎜ ⎟ we mean is Full-WENO may quite different from SL/EL- ρ(e u + e v + e w)u + e p(u) x y z x ⎜ ⎟ ⎜ ⎟ WENO (Huang et al., 2016; Huang & Arbogast, 2017), + ρ(e u + e v + e w)v + e p(u) = 0, x y z y ⎜ ⎟ ⎝ ⎠ especially for systems. ρ(e u + e v + e w)w + e p(u) x y z z Note 4: In theory, with same computational condition, (e u + e v + e w)(E + p(u)) x y z the computing speed of Full-WENO will be s times faster (e = ξ, η, ζ). (22) than same order WENO with s-stage RK method. For example, for six-stage RK5 and nine-stage RK7 (Butcher, Equation (1) and the last equation of Equation (22) is 2016) (see Appendix B), the computing speed of Full- absolutely thesameformexceptthatthe number of com- WENO5willbe6timesfasterthanWENO5 + RK5, and ponents is different. So it is easy to generalize all for- Full-WENO7 will be 9 faster than WENO7 + RK7. mulas of schemes (Full-WENO5 and Full-WENO7) for Equation (22) from Equation (1). For example, the gen- eral expression of numerical u fl x for conservative scheme 4. Multi-dimensional cases in 3D can be written as follows For three-dimensional Euler equations in curvilin- e e n+1 n τ ˜ ˜ u = u − (f 1 − f 1 ), e = ξ, η, ζ , j h ear coordinates, we use dimensional splitting method j+ j - 2 2 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 11 Figure 1. Accuracy order test, linear equation. (A) 5-th order schemes (B) 7-th order schemes. 5.1.1. Test 1 Accuracy order test, linear equation k = 1, 2, 3, 4, 5, This is an accuracy order test of linear equation, initial e,k e,k e,k ∗e,k value is given as follows. f 1 = R (λ L u ¯ − ϕ ), j+ j+ k=1 u + u = 0, t = 2, t x n n e,k (24) v −v (e−λ τ) u (x) = sin(πx), x ∈ [−1, 1]. j+ 0 e,k e,k L u ¯ ≡ L . (23) e,k λ τ j+ 5.1.2. Test 2 Accuracy order test, Burgers equation This is an accuracy order test of nonlinear equation, initial value is given as follows. More detailed information on three-dimensional cases can be found in our former work (Zhou & Dong, 2021, 2 u + = 0, t = 0.15, 2022). (25) u (x) = 0.5 + sin(πx), x ∈ [0, 2]. 5. Numerical experiments As shown in Figures 1 and 2,bothFull-WENO and WENO + RK almost achieve designed accuracy order. This section we do numerical experiments for Full- Full-WENO reaches similar accuracy order compared WENO, which contains accuracy order tests, scalar with the same order WENO +RK,however,withsmaller equation tests, Euler equation tests, and sonic point test. errors. While WENO + RK3 only shows about third For all tests, we compare our new schemes with fre- order under CFL = 0.5 duetothe spacediscretedoesnot quently used WENO-RK schemes. If not specifically dominate the accuracy order at this CFL number. In test 1 stated WENO-RK uses TVD-RK3 (Jiang & Shu, 1996) Full-WENO7 and WENO7 + RK7 only reach about sixth (see Appendix B) and Local Lax-Friedrichs (LLF) ux fl orderand they will reachabout eighth orderifwecon- (Svenn Tveit, 2011). All numerical simulations are cal- tinue to ren fi e the grids. After further research, we found culatedbyour owncodes developedinFortran95 with this phenomenon may cause by the positive real number Microsoft Visual Studio 2013 and no specific data is used. in smoothness indicator, which has been discussed in reference (Henrick et al., 2005). 5.1. Tests of accuracy order 5.1.3. Test 3 Point-by-point accuracy order (PbP This subsection we test accuracy order of new scheme, order) which contains linear equation, Burgers equation, and Here we construct a new method for testing accuracy point-by-point accuracy order tests. order. As the name suggests, point-by-point accuracy 12 H. DONG ET AL. Figure 2. Accuracy order test, Burgers equation. (A) 5-th order schemes (B) 7-th order schemes. Table 1. Accuracy order test, theoretical error of schemes. Theoretical error R (a = 1, T = 2, v = CFL) CFL = 0.1 CFL = 0.5 CFL = 0.9 CFL = 1.0 2 2 2 (1−ν )(2 −ν )(3−ν) 5 5 5 5 5 Full-WENO5 R = (aT| |2π )h 19.475 h 11.954 h 2.164 h 0 Full - WENO5 6! 2 2 2 2 2 (1−ν )(2 −ν )(3 −ν )(4−ν) 7 7 7 7 7 Full-WENO7 R = (aT| |2π )h 41.497 h 25.808 h 4.611 h 0 Full - WENO7 8! 2 ×3 5 5 5 WENO5 R = (aT| |2π )h 20.401 h WENO5 6! 2 2 2 ×3 ×4 7 7 7 WENO7 R = (aT| |2π )h 43.147 h WENO7 8! orderistoreckonthe accuracy orderatevery point without the consideration of the error origin from RK by applying the theoretical error and accuracy. This method. When CFL→1 the theoretical error of Full- method can visually locate positions of order loss and is WENO tends to be zero, and when CFL→0ittends to helpful for researchers checking and analyzing bugs of be same as semi-discrete WENO (for nonlinear cases algorithms. It’s meaningful for research of smoothness Full-WENOneedstoconsiderthe errorof ν = λτ/h). indicators. Consider following linear equation. Figure 3 shows the point-by-point accuracy order, where optimal value is calculated directly by optimal weight u + au = 0, t ≤ T, t x without smoothness indicator. Figure 3 also tells that (26) a = 1, T = 2. Full-WENO achieves similar accuracy order compared with same order WENO + RK, however, WENO + RK3 point-by-point accuracy order can be given as only achieves third to fourth order. And we can also nd, fi it may introduce new errors after applying smoothness R(x) q(x) = ln / ln h. (27) C(x) indicator and thus trigger an overall accuracy reduction, which means there is still room for improvement in this (q+1) |u (x)| (q+1) Where C(x) = Cu = C , and theoretical smoothness indicator. The accuracy order test method in (q+1) |u | error R = Ch has been given in Table 1, C is theo- Test 1 may mask this problem due to the division of two retical error coefficient, q is theoretical accuracy order, errors at different grids, and nall fi y get designed accuracy. (q + 1) (q+1) |u (x)| is q-th derivative of initial value, |u | is theaverage valueof q-th derivative, h is space step. Here we test an inn fi itely smooth case, initial value can 5.2. Scalar equation be foundinEquation(24),and forthiscasewehave 5.2.1. Test 4 Linear scalar equation with multiple q+1 π cos(πx) , q = 2k − 1, (q+1) 2π extremes u = k = 1, 2, 3 ... q+1 π sin(πx) This test is composited by a series of smooth and , q = 2k, 2π Table 1 shows that, for linear cases, the theoretical unsmooth functions, which contains a Gaussian, a error of WENO + RK is not relevant to CFL number square, a triangle, and a semi-ellipse. It is widely used ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 13 Figure 3. Accuracy order test, point-by-point accuracy. (A) 5-th order schemes (B) 7-th order schemes. (G(x, β, z − δ) + G(x, β, z + δ) + 4G(x, β, z)), −0.8 ≤ x ≤−0.6, ⎪ 6 1, −0.4 ≤ x ≤−0.2, u (x) = 1 −|10(x − 0.1)|,0 ≤ x ≤ 0.2, ⎪ (F(x, α, a − δ) + F(x, α, a + δ) + 4F(x, α, a)), 0.4 ≤ x ≤ 0.6, (28) 0, else, −β(x−z) 2 G = e , F(x, α, a) = max(1 − α (x − a) ,0), z =−0.7, δ = 0.005, β = (log 2)/(36δ ), a = 0.5, α = 10. The result of Figure 4 is solved with 200 points and tells that Full-WENO is also equipped with an excel- t = 8, which shows the quite different solution properties lent low dissipative property in nonlinear cases, which is between these two schemes. Combining Table 1,wecan mainly attributed to the consistent high-order spatial and know theresolutionofFull-WENO forthislinearcase temporal accuracy. tend to be similar with semi-discrete WENO + RK when CFL→0and to be exactsolutionwhenCFL→1. On the 5.3. One-dimensional Euler equations contrary, due to RK3 method needs a small CFL number to maintain its robustness and accuracy, the resolution of 5.3.1. Test 6 Sod shock tube problem This is a typical Riemann problem for 1D Euler equa- WENO + RK3 performs worse with the increasing CFL number tions. Initial value is given as follows. (1, 0, 1),0 ≤ x ≤ 0.5, (ρ, u, p) = (29) (0.125, 0, 0.1), 0.5 ≤ x ≤ 1. 5.2.2. Test 5 Burgers equation This is a nonlinear case, the initial value has been given The solutions at t = 0.2 with 200 points, CFL = 1for in Equation (25). Forty points and period boundary Full-WENO, CFL = 0.4 for WENO + RK3 and recon- are used, and we output the solution at t = 0.5 and structed eigenvectors are plotted in Figure 6.Itcan be t = 20 respectively. And we set CFL = 1forFull-WENO, seen that, owing to the high order u fl x reconstruction CFL = 0.4 for WENO + RK3. Figure 5 shows that Full- for rarefaction waves and linear field, Full-WENO resolve WENO performs better at the discontinuity and still the rarefaction waves and contract discontinuity much keeps a high resolution after a long-time evolution, while better than WENO + RK3. Also, owing to Roe means are WENO + RK3 obviously loses its resolution. This test applied for the compression waves that may evolve into 14 H. DONG ET AL. Figure 4. Linear scalar equation, multiple extremes case. (A) Full-WENO5, overall view (B) Full-WENO5, local view (C) WENO5 + RK3, overall view (D) WENO5 + RK3, local view (E) Full-WENO7, overall view (F) Full-WENO7, local view (G) WENO7 + RK3, overall view (H) WENO7 + RK3, local view. ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 15 Figure 5. Nonlinear scalar equation, Burgers equation. (A) 5-th order schemes, t = 0.5 (B) 7-th order schemes, t = 0.5 (A) 5-th order schemes, t = 20 (B) 7-th order schemes, t = 20. shocks, the performance of Full-WENO at the shock is 10000 points. It is shown in Figure 7 that Full-WENO also better than WENO + RK3 with LLF u fl x. resolves the density profile much better than same order WENO + RK3, especially near the valley x = 0.75 and 5.3.2. Test 7 Blast-wave problem the right peak x = 0.78, which means that Full-WENO We now consider the interaction of two blast waves. exhibits less dissipation than WENO + RK3. Initial value is given as follows. 5.3.3. Test 8 Shu-Osher problem (1, 0, 10 ),0 ≤ x < 0.1, This is a typical example for testing the performance −2 (ρ, u, p) = (1, 0, 10 ), 0.1 ≤ x < 0.9, (30) ⎩ of high-order schemes when the solution contains both (1, 0, 10 ), 0.9 ≤ x < 1. shocks and complex smooth region structures. In this A reflective boundary condition is imposed at both case, a one-dimensional Mach 3 shock wave interacts ends. The simulation is performed with 500 points until with a perturbed density eld fi generating both small- final time t = 0.038. The reference solution is obtained scale structures and discontinuities, hence it is selected to by second-order TVD scheme (Harten, 1983)with validate shock-capturing and wave-resolution capability. 16 H. DONG ET AL. Figure 6. 1D Euler equations, Sod shock tube problem. (A) 5-th order schemes, overall view (B) 5-th order schemes, local view (C) 7-th order schemes, overall view (D) 7-th order schemes, local view. Initial value is given as follows. test well confirms that Full-WENO is less dissipation than WENO + RK3, which mainly attributes to its high-order (3.857, 2.629, 10.333), −5 ≤ x < −4, accuracy in time evolution. (ρ, u, p) = (1 + 0.2 sin(5x),0,1), −4 ≤ x < 5. (31) 5.4. Two-dimensional Euler equations Here we solve the case with 200 and 400 grids 5.4.1. Test 9 Two-dimensional Riemann problem respectively, and we set CFL = 1for Full-WENO, This is a Riemann problem with interaction of planar CFL = 0.4 for WENO + RK3. The reference solution is shocks. Initial value is given as follows. also obtained by second-order TVD scheme with 10,000 points. Figure 8 is the solution output at t = 1.8, which tells that Full-WENO performs much better than same order WENO +RK3bothatcoarseand nfi egrids.This (p, u, v, p) ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 17 Figure 7. 1D Euler equations, Blast-Wave problem. (A) 5-th order schemes (B) 7-th order schemes. Where indicates vortex strength, α controls the decay (0.138, 1.206, 1.206, 0.029), x < 0.8, y < 0.8, rate of vortex, r is the critical radius to control the max- (0.5323, 0, 1.206, 0.3), x > 0.8, y < 0.8, c imum vortex strength. Here we set = 0.3, r = 0.05, (1.5, 0, 0, 1.5), x > 0.8, y > 0.8, c α = 0.204. The computational domain is taken as (0.5323, 1.206, 0, 0.3), x < 0.8, y > 0.8. (x, y)∈[0, 2] × [0, 1], and the mesh of 200 × 80 is (32) used. We set CFL = 1for Full-WENO, CFL = 0.4 for We solvethe 2D Eulerequations in acomputational WENO + RK3. Figures 10 and 11 are the solution output domain of (x, y)∈[0,1] × [0,1] with the mesh resolu- at t = 0.6. tion 400 × 400 until nal fi time t = 0.8. We set CFL = 1 Figure 10 is the comparison of the density along for Full-WENO, CFL = 0.4 for WENO + RK3. Figure 9 thecenterlineof y = 0.5, which shows Full-WENO shows that Full-WENO resolves much better than same has less numerical dissipation according to the perfor- order WENO + RK3 in predicting the small-scale struc- mance of moving vortex. And Full-WENO also shows tures its excellent capturing capacity in Figures 10 and 11,for which the shock of Full-WENO is much sharper than 5.4.2. Test 10 Shock vortex interaction problem WENO + RK3. It is mainly because the Roe means inlaid This model problem describes the interaction between in u fl x linearization is friendly to shock. However, LLF a stationary shock and a vortex. At the initial moment, flux applied in WENO + RK3 may introduce some dissi- a stationary Mach 1.1 shock is positioned at x = 0.5 pation. Moreover, if use Roe flux for WENO + RK3, some and normal to the x-axis. Its left state is (p, u, v, p) = entropy xfi may be needed to maintain its robustness and (1, γ,0,1), and its right state can be obtained through correctness which may also introduce some dissipation, Hugoniot relation. A small vortex is superposed to the see analysis in section 5.6. flow left to the shock and centers at ( x , y ) = (0.25, 0.5). c c The state of vortex can be described as a perturbation 5.4.3. Test 11 Rayleigh-Taylor instability problem to velocity (u, v), temperature (T = p/ρ)and entropy This problem illustrates the flow caused by the instability (S = ln(p/ρ )) of the mean flow, then we can denote it in domain (x, y)∈ [0, 0.25] × [0, 1]. Initial value is given by the tilde values. as follows. α(1−τ ) u ˜ = ετe sin θ, α(1−τ ) v ˜ =−ετe cos θ, (ρ, u, v, p) 2 2α(1−τ ) (γ −1)ε e T =− , (33) 4αγ (2, 0, −0.025a · cos(8πx),2y + 1),0 ≤ y < 0.5, S = 0, (1, 0, −0.025a · cos(8πx), y + 1.5), 0.5 ≤ y ≤ 1. 2 2 (34) τ = r/r , r = (x − x ) + (y − y ) . c c c 18 H. DONG ET AL. Figure 8. 1D Euler equations, Shu-Osher problem. (A) 5-th order schemes, 200 points (B) 7-th order schemes, 200 points (C) 5-th order schemes, 400 points (D) 7-th order schemes, 400 points. andspace of Full-WENOwillleadtolessnumerical Where, a = γ p/ρ γ = is the sound speed. The dissipation. Moreover, a more relaxed CFL lead to fewer boundary conditions are set as follows: the initial con- time steps for Full-WENO, which also helps to reduce the dition at the bottom boundary is (ρ, u, v, p) = (2, 0, numerical dissipation. 0, 1); while at the top boundary (ρ, u, v, p) = (1, 0, 0, 2.5) is assigned; at the left and right boundaries, the reflecting boundary conditions are used. We solve the two-dimensional Euler equations by adding the source 5.4.4. Test 12 Double Mach reflection problem term (0, 0, ρ, ρv). Mesh 240 × 960 is used in this test, This is a classic test for investigating high-resolution and we set CFL = 1for Full-WENO, CFL = 0.4 for schemes. We solve the 2D Euler equations in a compu- WENO + RK3. We output the result at t = 1.95. As tational domain of (x, y)∈[0, 4] × [0, 1], and the mesh of described in Figure 12,Full-WENO hasresolvedmuch 1600 × 400 is adopted. Here we also set CFL = 1forFull- richer vortical structures than same order WENO. This WENO, CFL = 0.4 for WENO + RK3. At the bottom of is because the uniform high order accuracy in time computational domain, it is a reflecting wall. Initially, a ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 19 Figure 9. 2D Euler equations, density distribution of 2D Riemann problem, 30 contours from 0.12 to 1.76. (A) Full-WENO5 (B) WENO5 (C) Full-WENO7 (D) WENO7. strong shockofMach10makesa60° anglewiththe bot- 5.5. CPU time test tom wall which begins at x = 1/6 and extends to the top This section we test the CPU time (in sec.) of 1D and 2D of the domain at y = 0. Ahead of the shock, the initial tests respectively. All tests are obtained by same worksta- condition is ρ = 1.4, p = 1.0, u = 0, v = 0, and the exact tion (AMD Ryzen Threadripper 3790X 32-Core Proces- post-shock conditionisimposed by Hugoniot relation. sor, 3.69 GHz) and same working environment. As shown in Figure 13, Full-WENO shows its merits of high resolution and low dissipation by resolving much more small structures compared with WENO + RK3. 5.5.1. Test 13 CPU time test, one-dimensional Euler We also provide the original single entropy condition equations whichisguidedonlybyvelocity. In Figure 13,wecan Here we compare the efficiency of fully discrete Full- n fi d the resolution of double entropy condition improves WENO and semi-discrete WENO +RK.Herewechoose much compared to that of single entropy condition (the the Sod shock tube, initial value sees Equation (29). In reason has been explained in section 2.2). theory, the computing speed of Full-WENO should be 20 H. DONG ET AL. Figure 10. 2D Euler equations, shock vortex interaction, density at y = 0.5. (A) 5-th order schemes (B) 7-th order schemes. Table 2. CPU time test, Two-dimensional Riemann problem. s/CFL (CFL = (CFL of WENO + RK)/ semi/fully semi/fully (CFL of Full-WENO)) times faster than same order Two-dimensional Riemann problem, grid number = 400× 400, t = 0.8, 28 threads WENO with s-stage RK method. As described in Figure scheme CFL CPU time (Rate) CFL CPU time (Rate) 14, for 3-stage RK3, 6 stage-RK5, and 9-stage RK7, the r = 3 CPU time tests all basically meet this law. Taking RK3 Full-WENO5 0.4 74.7 (2.44) 1.0 30.6 (1.00) as an example: (1) With same CFL number, the com- WENO5 + RK3 0.4 206.1 (6.74) 1.0 84.0 (2.75) r = 4 puting speed of Full-WENO achieve about 2.8 times Full-WENO7 0.4 88.4 (2.44) 1.0 36.2 (1.00) compared with same order WENO + RK3; (2) Consid- WENO7 + RK3 0.4 244.1 (6.75) 1.0 100.8 (2.79) ering WENO + RK3 requires a small CFL number for its Table 3. CPU time test, Shock vortex interaction. resolution, with different CFL numbers, the computing speed of Full-WENO (CFL = 1) achieves about 6.8 times Shock vortex interaction, grid number = 200× 80, t = 0.6, 1 thread scheme CFL CPU time (Rate) CFL CPU time (Rate) compared with WENO + RK3 (CFL = 0.4). r = 3 More specicfi ally, here we extract a data from Figure 14 Full-WENO5 0.4 8.7 (2.23) 1.0 3.9 (1.00) (800 points) to conclude the computing speed compari- WENO5 + RK3 0.4 24.9 (6.38) 1.0 11.2 (2.87) r = 4 son in same order. Figure 15 tells that compared to the Full-WENO7 0.4 11.2 (2.15) 1.0 5.2 (1.00) same time-space accuracy WENO + RK, Full-WENO WENO7 + RK3 0.4 32.8 (6.31) 1.0 15.0 (2.92) is extremely equipped with competition in efficiency. The computing speed of Full-WENO5 can reach about 5.5 times as much as WENO5 + RK5 and Full-WENO7 WENO + RK3 under CFL = 1, and achieve about 6.8 canreach about8timesasmuchasWENO7 + RK7 times if WENO + RK3 uses CFL = 0.4 with considera- under same computing conditions, which also basically tion of robustness and resolution. Because the entropy conforms to the theoretical speed (namely 6 and 9 condition applies different strategies for compression times, respectively). Moreover, the storage memory of wavesand rarefactionwaves (lesscomputing burden for RK method is also more than s times compared to Full- compression waves), some differences may happen in WENO. CPU time rate. (Note: We do not consider the CPU time of source terms in Rayleigh-Taylor instability problem) 5.5.2. Test 14 CPU time test, two-dimensional Euler equations 5.6. Sonic point test The CPU time test of all 2D cases has been given in the corresponding test under section 5.4 (in sec). From For schemes do not satisfy the entropy condition may Tables 2–5, we can obviously n fi d the computing speed render anon-physicalsolutionatthe rarefactionwave of Full-WENO achieve about 2.8 times as much as with a sonic point, such as WENO with Roe ux fl (just ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 21 Figure 11. 2D Euler equations, density distribution of shock vortex interaction, 30 contours from 1 to 1.22. (A) Full-WENO5 (B) WENO5 (C) Full-WENO7 (D) WENO7. Table 4. CPU time test, Rayleigh-Taylor instability problem. Table 5. CPU time test, Double Mach reﬂection problem. Rayleigh-Taylor instability problem, grid number = 240× 960, Double Mach reﬂection problem, grid number = 1600× 400, t = 0.2, t = 1.95, 28 threads 28 threads scheme CFL CPU time (Rate) CFL CPU time (Rate) scheme CFL CPU time (Rate) CFL CPU time (Rate) r = 3 r = 3 Full-WENO5 0.4 279.1 (2.42) 1.0 115.1 (1.00) Full-WENO5 0.4 297.4 (2.42) 1.0 123.9 (1.00) WENO5 + RK3 0.4 792.1 (6.88) 1.0 330.2 (2.87) WENO5 + RK3 0.4 846.2 (7.07) 1.0 356.8 (2.88) r = 4 r = 4 Full-WENO7 0.4 335.1 (2.41) 1.0 139.1 (1.00) Full-WENO7 0.4 383.0 (2.40) 1.0 159.6 (1.00) WENO7 + RK3 0.4 935.0 (6.72) 1.0 388.6 (2.79) WENO7 + RK3 0.4 1090.1 (6.83) 1.0 448.5 (2.81) like that Roe scheme does not satisfy entropy condi- tion). However, Full-WENO can ecffi iently avoid the 5.6.1. Test 15 Sonic point test, reversed shock problem non-physical solution by formula solution method con- We design areversedshock problemthatcontainsasonic structed with entropy condition linearization. pointatrarefaction.Fivehundred points areused, initial 22 H. DONG ET AL. Figure 12. 2D Euler equations, density distribution of Rayleigh-Taylor instability problem, 50 contours from 0.9 to 2.2. (A) Full-WENO5 (B) WENO5 (C) Full-WENO7 (D) WENO7. value is given as follows. high accuracy order, high resolution to discontinuities, and high speed of computation, when compared to tra- (5.4, 7/9,31/3), −0.5 ≤ x ≤ 0, (ρ, u, p) = (35) ditional WENO-RK. The accuracy order tests verify that (1.4, 3.0, 1.0),0 ≤ x ≤ 1.5. new schemes reach the designed accuracy order. Scalar In this test, we output the result at t = 0.3 and use the tests including multiple extremes test and nonlinear fifth-order scheme as an example. WENO5 + RK3 with test show that new scheme’s resolution to discontinu- Roe ux fl is set as the comparison, and is imposed with ities and extremes is obviously higher than WENO- an entropy xfi Q(λ)(Harten, 1983)( = an extra articfi ial RK. One-dimensional Euler equation tests including Sod viscosity at the sonic point) shock tube, Blast-wave, and Shu-Osher, prove that the solution formula method is feasible in u fl id flow sim- |λ|, |λ| >ε, ulations, and keeps the priority to WENO-RK. Two- Q(λ) ≡ 0 ≤ ε ≤ 1. (36) 2 2 λ +ε ,|λ|≤ ε, dimensional Euler equation tests including 2D-Reimann, 2ε Shock-vortex, RT-instabilities, and Double Mach reflec- As shown in Figure 16,WENO5-Roe hasalargeunphys- tion show more cases of high-quality results of new ical discontinuity and tends to be normal only if an schemes using double entropy conditions, such as cap- extra articfi ial viscosity is used. While Full-WENO5 has turing much more n fi e structures than WENO + RK. well resolved the solution owing to its entropy condition CPU time test shows that new scheme has a speed of solution formula without any articfi ial viscosity. computation many times faster than WENO-RK. Sonic point test shows that new scheme can effectively avoid 5.7. Summary for numerical experiments the unphysical discontinuity without any extra arti-fi Overall, the numerical tests show that Full-WENO is cial viscosity, thanks to entropy condition inlaid in u fl x equipped with the merits of high ecffi iency including linearization. ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 23 Figure 13. 2D Euler equations, density distribution of Double Mach reﬂection problem, 30 contours from 2 to 22. (A) Full-WENO5 (double entropy condition) (B) Full-WENO5 (single entropy condition) (C) WENO5 (D) Full-WENO7 (double entropy condition) (E) Full-WENO7 (single entropy condition) (F) WENO7. 24 H. DONG ET AL. Figure 14. CPU time test, one-dimensional Euler equations, Sod shock tube problem. (A) 5-th order schemes (B) 7-th order schemes. Figure 15. CPU time test, computing speed comparison in same order. 6. Conclusions The solution formula method is a construction method Figure 16. Sonic point test, reversed shock problem. for difference schemes which we are in favor of. It is equipped with following merits: clear philosophy, easy derivation, natural upwind, and consistent space-time accuracy achievable.Inthispaper,weconstruct aFull- all semi-discrete schemes propelled with RK method do WENO scheme by solution formula method combined not have this merit, which indicates that Full-WENO with some techniques of WENO reconstruction. The can reserve more accurate information of solutions. new scheme has following vfi e highlights: (1) One- (3) High resolution: Full-WENO resolves better than step and fully discrete: Full-WENO gets rid of RK same order WENO + RK for all CFL numbers. More method and achieves consistent high accuracy order importantly, the larger the CFL number, the more obvi- both in space and time. And Full-WENO can per- ous the performance gap will be; (4) High computing form robustly under CFL = 1; (2) The excellent perfor- speed: The computing speed of Full-WENO can reach mance along with CFL number: When CFL→1, Full- about s timesasmuchassemi-discrete WENO + s- WENO not only tends to be exact in linear cases, but stage RK under same computational condition. Specif- also to be more accurate in nonlinear cases. This is ically, Full-WENO5 is nearly 5 ∼6times faster than named the traveling wave solution property. However, WENO5 + RK5, Full-WENO7 is nearly 8 ∼9timesfaster ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 25 than WENO7 + RK7; (5) Double entropy condition lin- Dumbser, M. (2014). Arbitrary-Lagrangian–Eulerian ADER– WENO finite volume schemes with time-accurate local earization: Full-WENO uses the newly designed double time stepping for hyperbolic conservation laws. Computer entropy condition linearization guided by both velocity Methods in Applied Mechanics and Engineering, 280, 57–83. and pressure. Compared with single entropy condition https://doi.org/10.1016/j.cma.2014.07.019 only guided by velocity, the new method is more accu- Harten, A. (1983). High resolution schemes for hyperbolic rate and reliable. It remedies the accuracy loss caused conservation laws. Journal of Computational Physics, 49(3), 357–393. https://doi.org/10.1016/0021-9991(83)90136-5 by rough prediction of previous method (especially for Henrick, A. K., Aslam, T. D., & Powers, J. M. (2005). Mapped multi-dimensional problems using Strang split). More- weighted essentially non-oscillatory schemes: Achieving over, compared with WENO + RK which needs entropy optimal order near critical points. Journal of Computational xfi for Roe flux (or using LLF flux which has more Physics, 207(2), 542–567. https://doi.org/10.1016/j.jcp.2005. numerical viscosity) to avoid non-physical solutions, 01.023 Full-WENO embedded with entropy condition automat- Huang, C-S, & Arbogast, T. (2017). An eulerian–lagrangian weighted essentially nonoscillatory scheme for nonlinear ically avoids non-physical solutions without extra artifi- conservation laws. Numerical Methods for Partial Dieff ren- cial numerical viscosity. tial Equations, 33(3), 651–680. https://doi.org/10.1002/num. The solution formula method is valuable for engi- neering applications, particularly for long-time evolution Huang, C-S, Arbogast, T., & Hung, C-H. (2016). A semi- problems. The entropy condition linearization is a criti- Lagrangian finite dieff rence WENO scheme for scalar nonlinear conservation laws. Journal of Computational cal technique in solution formula method which largely Physics, 322(4), 559–585. https://doi.org/10.1016/j.jcp.2016. determines the resolution and robustness in nonlinear 06.027 situations. However, there are still some flaws in current Jiang, G., & Shu, C. W. (1996). Efficient implementation of work: (1) How to remove the influence of numerical dis- weighted ENO schemes. Journal of Computational Physics, turbances on the entropy condition selection which may 126(1), 202–228. https://doi.org/10.1006/jcph.1996.0130 result in a resolution loss due to incorrectly choose the Lax, P. (1971). Shock waves and entropy. In E. H. Zarantonello (Ed.), Contributions to nonlinear functional analysis (pp. Roe means; (2) How to exactly predict when the com- 603–634). Academic Press. https://doi.org/10.1016/B978-0- pression waves evolve into shocks (we can also apply the 12-775850-3.50018-2 high-order flux reconstruction for compression waves Lax, P., & Wendro,ff B. ( 1960). Systems of conservation laws. before they evolve into shocks), etc. By solving these, Communications on Pure and Applied Mathematics, 13, the quality and robustness of numerical solutions may 217–237. https://doi.org/10.1002/cpa.3160130205 Li, J., & Du, Z. (2016). A Two-stage fourth order time-accurate be lifted to a new level. Therefore, in the future, we plan discretization for Lax-wendroff type flow solvers I. Hyper- to optimize the solution formula method from these two bolic conservation laws. SIAM Journal on Scientific Comput- aspects. Moreover, we also hope to develop this into a ing, 38(5), A3046–A3069. https://doi.org/10.1137/15M10- more efficient large time step scheme, which has been simply tried in (Zhou & Dong, 2022). If realized, it may Osher, S., & Harten, A. (1987). Uniformly high-order accu- greatly improve engineering ecffi iency. rate nonoscillatory schemes. I. SIAM Journal on Numerical Analysis, 24(2), 279–309. https://doi.org/10.1137/0724022 Note Qiu, J. (2007). WENO schemes with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations. Journal of Disclosure statement Computational and Applied Mathematics, 200(2), 591–605. https://doi.org/10.1016/j.cam.2006.01.022 No potential conflict of interest was reported by the author(s). Shen,Y., Liu, L.,&Yang,Y.(2014). Multistep weighted essentially non-oscillatory scheme. International Journal for References Numerical Methods in Fluids, 75(4), 231–249. https://doi.org/ 10.1002/fld.3889 Balsara, D. S., & Shu, C-W. (2000). Monotonicity preserving Shen,Y.,&Zha,G.(2014). Improvement of weighted essentially weighted essentially Non-oscillatory schemes with increas- non-oscillatory schemes near discontinuities. Computers ingly high order of accuracy. Journal of Computational &Fluids, 96,1–9. https://doi.org/10.1016/j.compu fl id.2014. Physics, 160(2), 405–452. https://doi.org/10.1006/jcph.2000. 02.010 Shu, C. W. (1997). Essentially non-oscillatory and weighted Butcher, J. C. (2016). Numerical methods for ordinary differ- essentially non-oscillatory schemes for hyperbolic conser- ential equations (3rd ed.). Chichester West Sussex United vation laws. In Advanced numerical approximation of non- Kingdom. linear hyperbolic equations (pp. 325–432). Springer Berlin Davis, S. F. (1987). A simplified TVD finite difference scheme Heidelberg. via artificial viscosity. SIAM Journal on Scientific and Statisti- Sohn, S.-I. (2005). A new TVD-MUSCL scheme for hyperbolic cal Computing, 8(1), 1–18. https://doi.org/10.1137/0908002 conservation laws. Computers & Mathematics with Appli- Dong, H., Lidong, Z., & Chun-Hian, L. (2002). High order dis- cations, 50(1-2), 231–248. https://doi.org/10.1016/j.camwa. continuities decomposition entropy condition schemes for 2004.10.047 Euler equations. CFD Journal, 10(4), 448–457. 26 H. DONG ET AL. Strang, G. (1968). On the construction and comparison of dif- Methods in Applied Mechanics and Engineering, 382, 113853. ference schemes. SIAM Journal on Numerical Analysis, 5(3), https://doi.org/10.1016/j.cma.2021.113853 506–517. https://doi.org/10.1137/0705041 Zhao, J., Özgen-Xian, I., Liang, D., Wang, T., & Hinkelmann, R. Svenn Tveit. (2011). Numerical methods for conservation laws (2019). An improved multislope MUSCL scheme for solv- with a discontinuous u fl x function [Master]. ing shallow water equations on unstructured grids. Com- Sweby, P. K. (1984). High resolution schemes using u fl x limiters puters & Mathematics with Applications, 77(2), 576–596. for hyperbolic conservation laws. SIAM Journal on Numeri- https://doi.org/10.1016/j.camwa.2018.09.059 cal Analysis, 21(5), 995–1011. https://doi.org/10.1137/0721- Zhou, T., & Dong, H. T. (2021). A fourth-order entropy con- 062 dition scheme for systems of hyperbolic conservation laws. Titarev, V. A., & Toro, E. F. (2002). Ader: Arbitrary high Engineering Applications of Computational Fluid Mechanics, order Godunov approach. Journal of Scientific Comput- 15(1), 1259–1281. https://doi.org/10.1080/19942060.2021. ing, 17(1/4), 609–618. https://doi.org/10.1023/A:101512681 1955010 4947 Zhou, T., & Dong, H. T. (2022). A sixth order entropy condi- Toro,E.F.(2009). Riemann solvers and numerical methods for tion scheme for compressible flow. Computers & Fluids, 243, fluid dynamics (3rd ed.). Springer. 105514. https://doi.org/10.1016/j.compu fl id.2022.105514 Toro,E.F., &Hidalgo,A.(2009). ADER finitevolumeschemes Zorío, D., Baeza, A., & Mulet, P. (2017). An approxi- for nonlinear reaction–diffusion equations. Applied Numer- mate Lax–wendroff-type procedure for high order accurate ical Mathematics, 59(1), 73–100. https://doi.org/10.1016/j. schemes for hyperbolic conservation laws. Journal of Sci- apnum.2007.12.001 entific Computing , 71(1), 246–273. https://doi.org/10.1007/ Wang,Z., Zhu, J.,Yang, Y.,&Zhao,N.(2021). A new s10915-016-0298-2 fifth-order alternative finite difference multi-resolution WENO scheme for solving compressible flow. Computer ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 27 Appendices Appendix A: Discretization process of Newton Interpolation in Full-WENO We give the detailed Newton interpolation process for both initial value reconstruction and ux fl reconstruction of solution formula method.ConsiderFull-WENO5, forinitial valuereconstructionincase a ≥ 0, there are three third-order Newton interpolations of υ at [x , x ]withsix data around it,the middle oneisasfollows 1 1 j− j+ 2 2 υ −υ υ −υ 3 1 1 1 j+ j+ j+ j− 2 2 2 2 x −x x −x υ −υ 3 1 1 1 1 1 j+ j+ j+ j− j+ j− n 2 2 2 2 2 2 υ(x) = υ + (x − x ) + (x − x )(x − x ) 1 1 1 1 x −x 1 1 j+ j− j+ j+ x − x 2 3 1 2 2 j+ j− j+ j− 2 2 2 2 υ −υ υ −υ υ −υ υ −υ 3 1 1 1 1 1 1 3 j+ j+ j+ j− j+ j− j− j− 2 2 2 2 2 2 2 2 − − x −x x −x x −x x −x 3 1 1 1 1 1 1 3 j+ j+ j+ j− j+ j− j− j− 2 2 2 2 2 2 2 2 x −x x −x 3 1 1 3 j+ j− j+ j− 2 2 2 2 + × (x − x )(x − x )(x − x ) 3 1 1 x −x j− j− j+ 3 3 2 2 2 j+ j− 2 2 u −u u −u j+1 j j j−1 u −u n j+1 j 2h 2h = υ + u (x − x ) + (x − x )(x − x ) + (x − x )(x − x )(x − x ) 1 1 1 3 1 1 1 2h j+ j− j+ j− j− j+ j+ 3h 2 2 2 2 2 2 j− = υ + u (x − x ) + (x − x )(x − x ) + (x − x )(x − x )(x − x ) (A1) j 1 1 1 3 1 1 1 2 2h 3!h j+ j− j+ j− j− j+ j+ 2 2 2 2 2 2 We write the three Newton interpolations in square brackets and deduce the expression of u as follows j+ ⎡ ⎤ j− j−1 n 2 υ + u x − x + x − x x − x + x − x x − x x − x ⎢ 1 1 1 3 1 1 ⎥ j 2 1 2h 3!h j+ j− j+ j− j− j+ ⎢ j+ ⎥ 2 2 2 2 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ j− ⎢ n 2 ⎥ υ(x) = , υ + u x − x + x − x x − x + x − x x − x x − x 1 1 1 3 1 1 ⎢ j 2 ⎥ 1 2h 3!h j+ j− j+ j− j− j+ j+ ⎢ 2 2 2 2 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ j+ n j+1 υ + u x − x + x − x x − x + x − x x − x x − x j 1 1 1 1 1 3 1 2h 3!h j+ j− j+ j− j+ j+ j+ 2 2 2 2 2 2 1 1 x = (j + )h, x = (j + − ν)h ⇒ x − x = (2 − ν)h, x − x = (1 − ν)h, x − x = (−ν)h, 1 3 1 1 2 2 j+ j− j− j+ 2 2 2 2 x − x = (−1 − ν)h, j+ ⎡ ⎤ j− j−1 n 2 3 υ + u (−ν)h + (1 − ν)(−ν)h + (2 − ν)(1 − ν)(−ν)h ⎢ j ⎥ 1 2 2h 3!h ⎢ j+ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ j− n+1 1 j ⎢ n 2 3 ⎥ υ ≡ υ((j + − ν)h) = , υ + u (−ν)h + (1 − ν)(−ν)h + (2 − ν)(1 − ν)(−ν)h 1 j 2 2 ⎢ 1 ⎥ 2h 3!h j+ j+ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ j+ j+1 n 2 2 3 υ + u (−ν)h + (1 − ν)(−ν)h + (1 − ν)(−ν)(−1 − ν)h j 2 2h 3!h j+ ⎡ ⎤ ⎡ ⎤ 2−ν 1−ν 1 2 u + + j− 1 j−1 j−1 2 ⎢ 3 2 ⎥ j− u + (1 − ν) + (2 − ν)(1 − ν) ⎢ 2 3! ⎥ ⎢ ⎥ n+1 υ − υ ⎢ ⎥ ⎢ ⎥ 1 1 ⎢ ⎥ ⎢ ⎥ j+ j+ 2 2 2−ν 1−ν ⎢ j− ⎥ ⎢ ⎥ u ¯ ≡ = j = u + + .(A2) 1 2 1 ⎢ ⎥ ⎢ j 3 2 ⎥ u + (1 − ν) + (2 − ν)(1 − ν) j+ j− νh 2 2 3! 2 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ 1 2 ⎣ ⎦ j+ 2 −1−ν 1−ν j+1 u + + j 1 u + (1 − ν) + (1 − ν)(−1 − ν) j j+1 3 2 2 3! j+ 2 28 H. DONG ET AL. For a ≥ 0, there is one fifth-order Newton interpolation at [ x , x ] with six data around it 1 j 1 j− j+ 2 2 j− n 2 υ(x) = υ + u (x − x ) + (x − x )(x − x ) + (x − x )(x − x )(x − x ) 1 1 1 3 1 1 j 2 2h 3!h j+ j− j+ j− j− j+ j+ 2 2 2 2 2 2 j+ + (x − x )(x − x )(x − x )(x − x ) + (x − x )(x − x )(x − x )(x − x )(x − x ), 3 1 1 3 5 3 1 1 3 3 4 4!h 5!h j− j− j+ j+ j− j− j− j+ j+ 2 2 2 2 2 2 2 2 2 1 1 x = (j + )h, x = (j + − ν)h ⇒ x − x = (3 − ν)h, x − x = (2 − ν)h, x − x = (1 − ν)h, 1 5 3 1 2 2 j+ j− j− j− 2 2 2 2 x − x = (−ν)h, x − x = (−1 − ν)h, x − x = (−2 − ν)h, 1 3 5 j+ j+ j+ 2 2 2 j− n+1 1 n j 2 2 3 υ ≡ υ((j + − ν)h) = υ + u (−ν)h + (1 − ν)(−ν)h + (2 − ν)(1 − ν)(−ν)h 1 2 1 2h 3!h j+ j+ 2 2 j+ 4 5 + (2 − ν)(1 − ν)(−ν)(−1 − ν)h + (3 − ν)(2 − ν)(1 − ν)(−ν)(−1 − ν)h , 3 4 4!h 5!h n n+1 υ −υ 1 1 j+ j+ 3−ν −1−ν 2−ν 1−ν 2 2 2 3 4 u ¯ ≡ = u + ( + ( + ( + ) ) ) . 1 1 j 1 j νh 5 4 3 2 j+ j− j− 2 2 (A3) For ux fl reconstruction in case a ≥ 0, we need the derivatives of three third-order Newton interpolations as follows ⎡ ⎤ j− j−1 d d d u x − x + x − x x − x + x − x x − x x − x j 1 1 1 3 1 1 ⎢ 2 ⎥ dx 2h dx dx 3!h j+ j− j+ j− j− j+ ⎢ 2 2 2 2 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ j− ⎥ d d d u(x) = υ (x) = , x ⎢ ⎥ u x − x + x − x x − x + x − x x − x x − x j 1 1 1 3 1 1 dx 2h dx dx ⎢ 3!h ⎥ j+ j− j+ j− j− j+ 2 2 2 2 2 2 ⎢ ⎥ ⎢ ⎥ ⎣ 2 ⎦ j+ j+1 d d d u x − x + x − x x − x + x − x x − x x − x j 1 1 1 2 1 1 3 dx 2h dx dx j+ j− j+ 3!h j− j+ j+ 2 2 2 2 2 2 ⎡ ⎤ j− j−1 d 2 d d 2 ⎢ u (−(−ν)) + (−(1 − ν)(−ν))h + (−(2 − ν)(1 − ν)(−ν))h ⎥ j 2 dv 2h dv 3!h dv ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ n+1 j− u ≡ u(j + − ν) = j , ⎢ d 2 d d 2 ⎥ u (−(−ν)) + (−(1 − ν)(−ν))h + (−(2 − ν)(1 − ν)(−ν))h j 2 j+ ⎢ ⎥ dv 2h dv 3!h dv ⎢ ⎥ ⎣ ⎦ j+ j+1 d 2 d d 2 u (−(−ν)) + (−(1 − ν)(−ν))h + (−(1 − ν)(−ν)(−1 − ν))h j 2 dv 2h dv 3!h dv ⎡ ⎤ d −(−v)(1−v) 2 d −(−v)(1−v)(2−v) u + + j 1 dv 2! j−1 dv 3! j− ⎢ ⎥ ⎢ ⎥ d −(−v)(1−v) d −(−v)(1−v)(2−v) ⎢ 2 ⎥ u + + = j .(A4) ⎢ j ⎥ dv 2! dv 3! j− ⎢ ⎥ ⎣ ⎦ d −(−v)(1−v) 2 d −(−1−v)(−v)(1−v) u + + j 1 dv 2! j+1 dv 3! j+ And the expansion form of ux fl reconstruction in Equation (19) (for Full-WENO5) can be given as follows ⎡ ⎤ −2v+1 2 3v −6v+2 ⎪ u + + j 1 ⎪ j−1 2! 3! j− ⎢ 2 ⎥ ⎢ −2v+1 2 3v −6v+2 ⎥ u + + ⎪ j 1 , a ≥ 0, ⎢ ⎥ 2! j 3! j− ⎡ ⎤ ⎪ ⎣ 2 ⎦ ∗ ⎪ ⎪ 2 −2v+1 3v −1 ⎪ 2 ⎨ u + + j+1 2! 3! j+ ⎢ ⎥ u ≡ ⎡ ⎤ (A5) ⎣ ⎦ 2 −2v−1 2 3v −1 u + + ⎪ j+1 1 2! 3! ∗ ⎪ j+ u ⎢ 2 ⎥ ⎢ −2v−1 2 3v +6v+2⎥ ⎪ u + + j+1 3 , a ≤ 0. ⎢ ⎥ ⎪ 2! j+1 3! j+ ⎣ 2 ⎦ ⎪ −2v−1 3v +6v+2 u + + j+1 j+2 2! 3! j+ 2 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 29 The expansion form of flux reconstruction in Equation (20) (for Full-WENO7) can be given as follows ⎧ ⎡ ⎤ 2 3 2 −2v+1 2 3v −6v+2 3 −4v +18v −22v+6 u + + + j 1 2! j−1 3! 3 4! ⎪ j− ⎢ j− ⎥ ⎪ 2 ⎪ ⎢ ⎥ 2 3 2 ⎪ −2v+1 3v −6v+2 3 −4v +6v −2v−2 ⎪ ⎢ ⎥ u + + + j 1 2! j 3! 1 4! ⎢ ⎥ ⎪ j− j− ⎪ 2 ⎢ 2 ⎥ , a ≥ 0, 2 3 2 ⎪ ⎢ −2v+1 3v −6v+2 −4v +6v −2v−2 ⎥ 2 3 u + + + ⎢ 1 ⎥ ⎪ j 2! j 3! 1 4! ⎡ ⎤ ⎪ j− ⎢ j+ ⎥ ∗ ⎪ 2 ⎪ 2 ⎪ ⎣ ⎦ 1 2 3 2 ⎪ −2v+1 3v −1 −4v −6v +2v+2 2 3 u + + + ⎢ ⎥ ⎪ j j+1 3 ∗ 2! 3! 4! ⎨ j+ u j+ ⎢ ⎥ 2 2 ⎢ ⎥ ≡ (A6) ⎡ ⎤ ⎢ ⎥ ∗ 2 3 2 ⎪ −2v−1 3v −1 −4v +6v −2v−2 u 2 3 ⎣ ⎦ ⎪ 3 u + + + ⎪ j+1 j 1 ⎪ 2! 3! 4! j+ ⎪ j− ⎢ ⎥ ∗ ⎪ 2 u ⎪ ⎢ ⎥ 2 3 2 4 ⎪ −2v−1 2 3v +6v+2 3 −4v −6v +2v+2 ⎪ ⎢ ⎥ u + + + j+1 3 j+1 1 ⎪ 2! 3! 4! ⎢ j+ ⎥ ⎪ j+ ⎢ 2 ⎥ ⎪ , a ≤ 0. 2 3 2 ⎪ ⎢ ⎥ −2v−1 2 3v +6v+2 3 −4v −6v +2v+2 ⎪ u + + + ⎢ j+1 3 ⎥ j+1 3 ⎪ 2! 3! 4! j+ ⎪ j+ ⎢ ⎥ ⎣ 2 3 2 ⎦ −2v−1 2 3v +6v+2 3 −4v −18v −22v−6 ⎩ u + + + j+1 3 2! j+2 3! 5 4! j+ j+ Appendix B: Runge-Kutta method used in WENO-RK TVD third-order three-stage RK method n+1 n 1 1 4 u = L(u), u = u + τ( L + L + L ) t 1 2 3 6 6 6 L ≡ L(u ), n (B1) ⎪ L ≡ L(u + τL ), n + 1 2 1 1 1 1 L ≡ L(u + τ( L + L )), n + 3 1 2 4 4 2 Nystrom fifth-order six-stage RK method n+1 n 23 125 −81 125 u = L(u), u = u + τ( L + L + L + L ) t 1 3 5 6 ⎪ 192 192 192 192 L ≡ L(u ), n n τ 1 ⎪ L ≡ L(u + L ), n + 2 1 ⎪ 3 3 4 6 2 L ≡ L(u + τ( L + L )), n + (B2) 3 1 2 25 25 5 n 1 −12 15 L ≡ L(u + τ( L + L + L )), n + 1 4 1 2 3 4 4 4 n 6 90 −50 8 2 ⎪ L ≡ L(u + τ( L + L + L + L )), n + 5 1 2 3 4 81 81 81 81 3 6 36 10 8 4 L ≡ L(u + τ( L + L + L + L )), n + 6 1 2 3 4 75 75 75 75 5 Butcher seventh-order nine-stage RK method n+1 n 32 1771561 243 16807 77 11 u = L(u), u = u + τ( L + L + L + L + L + L ) ⎪ t 4 5 6 7 8 9 105 6289920 2560 74880 1440 270 ⎪ L ≡ L(u ), n n τ 1 ⎪ L ≡ L(u + L ), n + 2 1 ⎪ 6 6 n τ 1 L ≡ L(u + L ), n + 3 2 ⎪ 3 3 1 3 1 L ≡ L(u + τ( L + L )), n + 4 1 3 8 8 2 (B3) 148 150 −56 2 L ≡ L(u + τ( L + L + L )), n + 5 1 3 4 ⎪ 1331 1331 1331 11 ⎪ −404 −170 4024 10648 2 L ≡ L(u + τ( L + L + L + L )), n + ⎪ 6 1 3 4 5 243 27 1701 1701 3 ⎪ n 2466 1242 −19176 −51909 1053 6 L ≡ L(u + τ( L + L + L + L + L )), n + ⎪ 7 1 3 4 5 6 2401 343 16807 16807 2401 7 ⎪ n 5 96 −1815 −405 49 L ≡ L(u + τ( L + L + L + L + L )), n ⎪ 8 1 4 5 6 7 154 539 20384 2464 1144 n −113 −195 32 29403 −729 1029 21 L ≡ L(u + τ( L + L + L + L + L + L + L )), n + 1 9 1 3 4 5 6 7 8 32 22 7 3584 512 1408 16 According to (Butcher, 2016): above order 4, it is no longer possible to obtain order s with just s stages. For order 5, six stages are required, and for order 7, nine stages are required. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Engineering Applications of Computational Fluid Mechanics Taylor & Francis http://www.deepdyve.com/lp/taylor-francis/fully-discrete-weno-with-double-entropy-condition-for-hyperbolic-QwWE7O4VlG

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Copyright: © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
ISSN: 1997-003X
eISSN: 1994-2060
DOI: 10.1080/19942060.2022.2145373
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Abstract

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 2023, VOL. 17, NO. 1, 2145373 https://doi.org/10.1080/19942060.2022.2145373 Fully discrete WENO with double entropy condition for hyperbolic conservation laws a b c Haitao Dong , Tong Zhou and Fujun Liu a b NLCFD, School of aeronautic science and engineering, Beihang University, Beijing, People’s Republic of China; School of aeronautic science and engineering, Beihang University, Beijing, People’s Republic of China; China Aerodynamic Research and Development Center, Mianyang, People’s Republic of China ABSTRACT ARTICLE HISTORY Received 15 July 2022 This paper put forward a new fully discrete scheme construction method – double entropy condition Accepted 30 October 2022 solution formula method. With that, we turn the state-of-the-art semi-discrete WENO + RK scheme into a fully discrete scheme, which is named as Full-WENO. A major difficulty of this work is that KEYWORDS we lack exact solution expressions for nonlinear equations in general cases. A feasible way we can solution formula method; go is to linearize equations and get quasi-exact solution formulas. The critical challenge is keeping fully discrete; full-WENO; both accuracy and efficiency in a scheme. Then, we get a class of new high-order schemes far better Euler equations; hyperbolic than traditional WENO schemes in the following aspects: (1) One-step to consistent high accuracy conservation laws; unsteady compressible ﬂow order in both space and time; (2) Resolution improves with the increasing CFL number; (3) Less CPU time and memory space, 1/s times of WENO with s-stage RK method in theory; (4) Excellent entropy condition satisfying property. Compared with our original work , the new method applies the more sophisticated WENO reconstruction and solves the resolution loss problems in multi-dimensional cases. The numerical tests show that the new scheme is equipped with the merits of high efficiency, high resolution and low dissipation, especially for long-time nonlinear problems. 1. Introduction Sweby, 1984), entropy condition for discontinuities (Lax, 1971), Riemann solvers (such as Roe, HLL, HLLC, etc.) Aerodynamic designs of vehicles strongly depend on (Toro, 2009), etc. With the pursuit of high-d fi elity numer- solutionsoffluidequations,however,exact solutions ical solutions to meet engineering needs, a large number can hardly be achieved due to nonlinearity of equations. of distinctive high-order schemes were constructed with At presentstage,solutions areusuallyobtainedintwo dieff rent ideas. By applying a rfi st-order TVD scheme ways: numerically through computational uid fl dynamics to equations with modified u fl xes, Harten constructed (CFD) and experimentally through wind tunnels. Wind a second-order TVD (Harten, 1983)scheme. By using tunnel experimentsare basedonrealfluids andthus limiters designed with the monotonicity preserving con- the measurement results are credible, but also costly and dition, VanLeer constructed the second-order- MUSCL dicffi ult to obtain complete information of flow. While scheme along the lines of the Godunov scheme, and now CFD can obtain more detailed flow information, there has been widely used in different elds fi (Sohn, 2005;Zhao are no guarantees that the results are reliable for all flow et al., 2019). Since the TVD condition is too strict, it is conditions. The two main factors that contribute to the not suitable for construction of higher-order schemes. credibility of CFD are algorithms and turbulence, and Some researchers also constructed higher-order schemes this paper is concerned with the former. which originated by Harten and developed by Osher From its earliest days, CFD has struggled with var- and Harten (1987)and Shu(1997) with a more relaxed ious flaws in its algorithms. One of the most serious essentially non-oscillatory (ENO) condition. Several flaws is the numerical oscillation of classical schemes years later, a compact semi-discrete weighted essentially caused by shocks. There have been ample research efforts non-oscillatory (WENO) scheme came out associated aimed at dealing with this problem, with partial success, with Runge-Kutta (RK) method which originated by LIU using techniques such as conservation forms, upwind and Osher and improved by Jiang and Shu (1996). Soon, difference, total variation diminishing (TVD) condi- it became the mainstream in high-order schemes due tion (Harten, 1983), limiters (Davis, 1987;Harten, 1983; CONTACT Fujun Liu liufujun2009@cardc.cn © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 H. DONG ET AL. to its excellent properties in shock-capturing. Under the to trace the characteristic line, which leads to a very impetus of Shu, etc., WENO schemes have been rapidly complex procedure compared with normal semi-discrete and widely used in many industries. Now, it has emerged WENO + RK. And for 1D Euler equations, their method avariety of WENO + RK schemes, such as MPWENO does not work very well due to there are three character- (Balsara & Shu, 2000), multistep WENO (Shen et al., istic lines, even computational cost is tripled (Huang & 2014;Shen&Zha, 2014), multi-resolution WENO (Wang Arbogast, 2017). et al., 2021) and in addition with Lax-Wendroff type Different from aforementioned methods, this paper WENO schemes (LW-WENO) (Li & Du, 2016;Qiu, constructs a one-step fully discrete scheme through solu- 2007;Zorío et al., 2017), ADER-WENO (Dumbser, 2014) tion formula method (Dong et al., 2002). The main point schemes united with Arbitrary high-order DErivative of solution formula method is to construct a quasi-exact Riemann (ADER) approaches (Titarev & Toro, 2002; solution formula for the partial differential equation and Toro & Hidalgo, 2009), and Semi-Lagrangian/Eulerian- then discretize it into a difference scheme. Since sys- Lagrangian (SL/EL) WENO schemes (Huang et al., 2016; tems of conservation laws are nonlinear hyperbolic type Huang & Arbogast, 2017). and their solutions may contain discontinuities, it is not Now we discuss the mentioned WENO schemes in easy to construct exact solution formulas for general ini- detail according to their time discretization method: tial values. So, first we integrate them once in space and (1) WENO + RK schemes are semi-discrete, which use obtain the Hamilton-Jacobi (HJ) equations. Although HJ multi-stage RK method to enhance time accuracy order equations are still nonlinear (in u fl x), their solutions are and avoid spurious oscillations. High-order RK meth- continuous and thus easier for discretization. Via lin- ods not only occupy a heavy burden in both computing earizing the u fl x of nonlinear HJ equations, we can obtain time and memory space, but also difficult to guarantee the quasi-exact solution formula. Applying Newton inter- TVD and robustness properties of the scheme. There- polation with limiters to the (quasi-) exact solution, Dong fore, researchers usually apply the third-order TVD-RK et al. (2002) constructed a second-order fully discrete (Jiang & Shu, 1996) method with a small CFL number. entropy condition (EC) scheme, and then also develop However, for some applications, such as numerical simu- it into a high-order version (Zhou & Dong, 2021, 2022). lation of compressible turbulence and wave propagation Since the solution formula of nonlinear equations con- problems involving long-time evolution it would be ben- taining discontinues is not easy to obtain, EC schemes eficial to use schemes which converge with higher order inlay the entropy condition of discontinuities into the flux both in time and space. (2) LW-WENO schemes are fully and thus get a quasi-exact solution after a flux lineariza- discrete, which realize time discretization by replacing all tion technique. This method can be generalized to sys- time derivatives with space derivatives through LW pro- tems, using discontinuity entropy conditions to get local cedure (Lax & Wendroff, 1960). Some can even improve quasi-exact solutions for systems (This method is easy their efficiency to twice of WENO + RK. Nevertheless, to extend to systems, simply after applying the entropy LW type schemes are extremely complex and not easy to condition of systems you can get the local quasi-exact implement in high-order situations, even need a multi- solution formula). In this paper, we leverage the solu- step strategy (Li & Du, 2016). (3) ADER-WENO schemes tion formula method and obtain a one-step fully discrete are one-step and fully discrete, which realize time dis- scheme by operating Newton interpolation with WENO cretization by LW procedure and simplified generalized construction, which is named as Full-WENO. Moreover, Riemann problem (GRP) solver. They decompose the we n fi d that it may result in some resolution loss or serve difficult problem into a sequence of m conventional Rie- oscillation in multi-dimensional problems when distin- mann problems and n fi ally achieve m-th accuracy order guishingthewaveevolutionofEulerequationsbythesin- (it can be arbitrary m-th accuracy order in theory). How- gle entropy condition adopted in EC schemes. The reason ever,thismay be alittlecostlywhenapplied to Euler is that it is only guided by velocity, however, the projected equations. Titarev and Toro (2002)toldthat, underthe velocity in multi-dimensional problems may be identified same CFL number, ADER schemes can be 2–3 times as different properties in different directions when apply- faster than WENO + RK for linear equations with con- ing Strang split technique (Strang, 1968). So, we design stantcoecffi ients, butonly50% faster for1DEuler equa- a more accurate and reliable double entropy condition, tions; (4) SL/EL-WENO schemes are fully discrete, which which applies selective ux fl reconstruction according to realize time discretization by tracing characteristic lines. both velocity and pressure. SL/EL-WENO can maintain its robustness in a relax CFL Main framework of this paper is as follows: the number, even nearly free from the limitation of CFL in second sectionintroducesgeneralconstructionsteps of cases of scalar nonlinear conservation laws (Huang et al., fully discrete schemes based on solution formula method; 2016). For scalar nonlinear cases, they apply RK method the third section proposes the concrete construction ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 3 processofFull-WENO;the fourth sectionextends new in this paper all indicates to the solution of HJ equation in scheme to three-dimensional Euler equations in curvi- Equation (2). Note: ‘ ≡ ’ means ‘define’ all over this paper. linear coordinates; the ffi th section is numerical exper- iments, includes accuracy order tests, tests of scalar 2.1. Construction of solution formula equation, one- and two-dimensional Euler equations, and sonic point test; the sixth section concludes. Dieff rentialequationisanalgebraic relation containing derivatives, while its solution is an algebraic relation con- taining no derivatives. Every part of scalar equation or 2. Construction of fully discrete schemes via matrix equations has same structure and is self-consistent solution formula method in algebraic relationships, which means one can directly This section considers the construction of schemes for construct their solution formulas. However, Euler equa- following one-dimensional Euler equations of perfect tions are vector systems, so we should use eigen-matrix gas. to transform vectors into scalars and get solution for- mulas. There are vfi e steps for constructing the solution ⎛ ⎞ ⎛ ⎞ ρ ρu formula of HJ systems: (1) scalarize (diagonalize) the sys- ⎝ ⎠ ⎝ ⎠ u + f(u) =0 ⇔ ρu + ρu + p =0, tem with eigenvector matrix; (2) linearize the ux fl via t x E u(E + p) first-order Taylor expansion or linear interpolation; (3) t x construct scalarized solution formula; (4) compose solu- ρu p p = (γ − 1)(E − ), c = γ.(1) tion formula for systems; (5) write general numerical u fl x 2 ρ by solution formula. Details of every step are as follows: Where ρ is density, u is velocity, p is pressure, E is total (1) Premultiply the HJ systems by local constant left energy, γ is specific heat ratio, c is sound speed. The eigenvectors Jacobian matrix of u fl x A = f (u)can be writteninto the form containing only velocity u and sound speed c, namely A(u,c). This matrix has complete left/right eigen- v + f(v ) = 0, t x n n vectors and eigenvalues, which can also be rewritten into v(x, t ) = v (x), the function of u and c: L(u,c), R(u,c), λ(u,c). diagonlization k k The Euler equations are actually conservation laws ↓ f(u) − −−−−−−→ ϕ (u) ≡ L f(u) ↓ (CL). First, we integrate the conservation laws once into k k L v + L f(v ) = 0, t x HJ equations. And then, we construct solution formulas (3) k n k n L v(x, t ) = L v (x). of HJ equations which will be used to obtain the numer- ical u fl x of conservative schemes. During this, there k k Where L is k-th row of matrix L, v and f are vectors, L v are two key techniques: (1) entropy condition lineariza- and L f are scalars, so Equation (3) shows the way from tion; (2) non-oscillation Newton interpolation methods. system (vector equations) to a group of scalar equations. Introducing the space integration of conserved variables, n n Note: the initial value v (x)startsfrom n-th timestep t we obtain HJ form of Euler equations. for the convenience of constructing numerical schemes. For one-dimensional Euler equations, k = 1, 2, 3, which u + f(u) = 0, t x (CL) means the k-th characteristic eld fi . n n u(x, t ) = u (x), (2) Rewrite the u fl x into linear form to linearize the udx ≡ v ↓↑ u ≡ v scalarized equations. v + f(v ) = 0, t x (HJ) (2) n n linearlization v(x, t ) = v (x). k k k k k ∗k ϕ (u) −→ λ L u−ϕ L v + L f(v ) =0, t x −→ k n k n L v(x, t ) =L v (x), Notice that the weak solution of CL systems is corre- k k k ∗k L v + λ L v − ϕ = 0, sponding to the continuous solution of HJ systems, and t x (4) k n k n L v(x, t ) = L v (x). it’s easier to get formulas of continuous solution. By using Newton interpolation, we can reconstruct solutions in k k k ∗k continuous fields from discrete data (numerical solu- Where ϕ (u) is a nonlinear function, λ L u-ϕ is a tion). This method greatly simpliefi s the derivation of linear function, so Equation (3) shows the way from non- schemes, which is easier for readers to understand the linear equation to linear equation. Note: L u is a scalar k ∗k essence of the algorithm. By the way, the solution formula variable, λ and ϕ are constants. 4 H. DONG ET AL. (3) The solution formula for the linearized scalarized yet. L u ¯ in above equations can be simplified into j+ equations can be given scalar form due to L are local constants. And for each k k k ∗k diagonalized solution formula L v + λ L v − ϕ =0, k theequations areall same,the k can be omitted. t x −→ k n k n L v(x, t ) =L v (x). n n υ − υ (x − λτ ) j+ j+ k n+1 k n k ∗k 2 k 2 × L v (x) = L v (x − λ τ) + τϕ .(5) L u ¯ = u ¯ ≡ j+ λτ j+ k k ∗k If we denote υ ≡ L v, a ≡ λ , f ∗≡ ϕ , the equation λτ of Equation (5) can be written as υ + aυ – f = 0, ↓ ν ≡ ↓ t x this linear equation has an exact solution expression as n n υ − υ n + 1 n 1 1 υ(x,t ) = υ (x-aτ)+τf ∗,justthe same as thesolu- j+ j+ −ν 2 2 n + 1 n u ¯ ≡ .(8) tion expression of Equation (5), where τ ≡ t -t is j+ νh time step size. Where the values with subscript j+1/2+ν are gener- ally not at the grid nodes, so we need an interpolation (4) The solution formula for systems can be composed method to construct the initial value υ (x). These are the by right eigenvectors whole idea of solution formula and general expression of n+1 1 1 n 1 ∗1 v (x) = R (L v (x − λ τ) + τϕ ) numerical u fl x. Then we just need to solve two key tech- 2 2 n 2 ∗2 niques: (1) Discontinuity entropy condition linearization + R (L v (x − λ τ) + τϕ ) techniquetodecidelocal constantsinsolutionformu- 3 3 n 3 ∗3 + R (L v (x − λ τ) + τϕ ).(6) las; (2) Non-oscillation Newton interpolation method to discrete numerical solution and to reconstruct the ini- Thesefoursteps abovecan reachthe solution formula tial value υ (x), and this paper we apply the thought of k k k ∗k expression of HJ systems. Where L , R , λ , ϕ are all WENO reconstruction. undetermined local constants, which will be given by dis- continuity entropy condition in next section. Note: R is 2.2. Double entropy condition linearization method the k-th column of matrix R. There exist global linearization methods for scalar non- (5) From solution formula Equation (6), we can obtain linear function, and thus exact solution formula can be the general expression of numerical u fl x for conser- constructed. However, the global linearization for non- vative scheme linear vector function cannot be achieved easily, so we try n n to construct quasi-exact formula by local linearization. v −v 1 1 j+ j− n 2 2 Local linearization is adopted at the interface of grid cells u ≡ n+1 n τ n n ˜ ˜ u = u − (f − f ) ←− [u , u ], which is a discontinuity formed by bilateral 1 1 j j+1 j j j+ j− 2 2 values. By the way, the linearization method for nonlin- n+1 n v = v − τf 1 ear vector is not unique, now we apply following double 1 1 j+ j+ j+ 2 2 entropy condition to determine the local constants (coef- n n +1 v −v ficients and constants of linear functions) in Equation 1 1 j+ j+ 2 2 f = , 1 (7): j+ u-entropy condition:Weapply velocity entropycondi- tion to obtain the local constants L, R, λ due to they n +1 k k n k ∗k v = R (L v (x − λ τ) + τϕ ), arematrixasawholethatdonot splitwithcharacter- j+ j+ k k k k=1 istic fields. Specifically, the local constants L , R , λ are constructed with the guide of bilateral velocity relation- ship u ≷u at arbitrary discontinuities [u , u ], so that L R L R we can scalarize the system. k k k ∗k f = R (λ L u ¯ − ϕ ), 1 1 j+ j+ √ √ 2 2 k=1 ρ u + ρ u L L R R ⎨ √ √ u > u L R ρ + ρ L R n n k v −v (x −λ τ) u ≡ , j+ j+ ⎩ ρ u +ρ u L L R R k k 2 u ≤ u L u ¯ ≡ L .(7) L R 1 ρ +ρ L R λ τ j+ 2 ⎧ √ √ ρ H + ρ H L L R R √ √ u > u L R ρ + ρ L R Note: Equation (7) is just the quasi-exact solution in a H ≡ , u +u L R H( ) u ≤ u numerical scheme form, it is not the n fi al discrete scheme L R 2 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 5 and pressure in union will be more accurate, especially for A ≡ A(u, c), multi-dimensional problems. The reason is the projected 2 L ≡ L(u, c), c = (γ − 1)(H − ) → (9) velocity may be identified as different properties (com- R ≡ R(u, c), pression wave or rarefaction wave) in different directions {λ }≡ λ(u, c). during applying Strang split technique (Strang, 1968), Concretely, we apply Roe means for u > u ,whilethe L R especially at the slip lines (contact discontinuities in 1D characteristic linestendtoconvergeintoashock. Andwe problems). Concretely, the scheme may apply Roe mean apply arithmetic means for u ≤ u , while the character- L R for compression waves in one direction and apply recon- n+1 isticlinetendstodivergeintorarefactionwaves.Withthis structed u in the other direction for rarefaction waves, j+ procedure, we can diagonalize the system. And here λ which may lead to a resolution loss or severe oscillation, will be used in following steps. such as that shown in section 5.4, Double Mach reflection p-entropy condition: We apply pressure entropy con- ∗k problem. With double entropy condition, we can more dition to obtain the local constants ϕ due to they can be precisely distinguish compression waves and rarefaction customized according to whether the characteristic elds fi are rarefaction waves or compression waves. Specicfi ally, waves. Moreover, we customize the reconstruction pro- ∗k the local constants ϕ are constructed with the guide posal for three efi lds of Euler equations. Due to the of pressure relationship p ≷ p at arbitrary discontinu- L/R second field of 1D Euler equations (second, fourth and ities [u , u ], so that we can linearize the scalarized u fl x. L R fifth fields in 3D Euler equations) is equipped with lin- Where p is an approximate middle pressure that can be ear features, we safely apply the high-order reconstructed obtained by discontinuity decomposition from bilateral n+1 u . status. j+ 2γ ⎛ ⎛ ⎞ ⎞ γ −1 u −u c c L R L R + + ⎜ ⎜ ⎟ ⎟ 2 γ −1 γ −1 p = max 0, , ⎝ ⎝ ⎠ ⎠ γ −1 γ −1 c 1 c 1 L 2γ R 2γ γ −1 p γ −1 p L R 3. Construction of full-WENO via solution u +u f(u )+f(u ) 1 L R L R formula method λ − , p > p , 2 2 ∗1 1 ϕ ≡ L , u +u u +u 1 L R L R λ − f( ), p ≤ p , 3.1. Initial value reconstruction 2 2 ∗2 2 2 u +u u +u L R L R ϕ ≡ L λ − f( ) , Using a Newton interpolation with WENO reconstruc- 2 2 u +u f(u )+f(u ) tion (Jiang & Shu, 1996), we construct the initial value 3 L R L R λ − , p > p , 2 2 ∗3 3 k n k n 1 ϕ ≡ L . (10) function L v (x)frominitial data L v , i = j + .And u +u u +u 3 L R L R λ − f( ), p ≤ p , n k n 2 2 for writing convenience, we denote υ (x) ≡ L v (x). (2r−1)-th Newton interpolation will occupy 2r−1grids, Concretely, we apply Roe means for p > p ,since these L/R consider the upwind condition and two numerical ux fl es, compression waves may evolve into shocks. Otherwise, the scheme will occupy 2r + 1 grids in total. According to we apply arithmetic means. This step is to linearize the WENO reconstruction, we weight several r-th stencils by scalarized equations. Note: Equations (7) and (8) and n n performing with smoothness indicators, which not only Equations (9) and (10) with [u , u ] ≡ [u , u ]make L R j j+1 helps to reach (2r−1)-th order at smooth regions but also up the so-called Solution Formula Method of this paper. can avoid the oscillation caused by high-order interpo- Above techniques which determine the local constants lation at risk regions. For example, a fifth-order stencil by predicting the discontinuities evolve into shocks or can be weighted by three third-order stencils, a seventh- rarefaction waves are named as entropy condition. Sim- order stencil can be weighted by four fourth-order sten- ilar to the statement in (Zhou & Dong, 2021, 2022), cils. Totally, this process will provide basic (2r−1)-th Equations (9) and (10) that provides basic second order orderinspace forFull-WENO scheme.Wegivethe spe- in temporal discretization for schemes in solution for- cicfi expressionsnow.(Note:wegivefifthand seventh mula method can be named as baseline double entropy schemes as examples here because it’s not easy to give the condition linearization. If we replace the arithmetic simple general expressions) means (u + u )/2 into a reconstructed high-order ver- L R n+1 sion u , it can provide higher-order nonlinear tempo- j+ ral accuracy order for scheme, details are shown in next section. 3.1.1. Initial value reconstruction for fifth order By the way, compared to the single entropy condition full-WENO (Full-WENO5) guided only by velocity applied in (Zhou & Dong, 2021, The fully discrete interpolation and weighted coefficients 2022), thedoubleentropy conditionguidedbyvelocity of Full-WENO5 can be given as. 6 H. DONG ET AL. 2 3 4 3−ν −1−ν 2−ν 1−ν ⎪ u + + + + , a ≥ 0, j 1 j 1 j weighted by 3 third order stencils ⎨ 5 4 3 2 j− j− 2 −→ u ¯ = j+ ⎪ ←− 2 3 4 −3−ν 1−ν −2−ν −1−ν u + + + + , a ≤ 0, finial fifth order stencils ⎩ j+1 j+1 3 j+1 5 4 3 2 j+ j+ (1)− (2)− (3)− − − − γ u + γ u + γ u ,a ≥ 0, 1 1 2 1 2 1 j+ j+ j+ 2 2 2 u ¯ = j+ (1)+ (2)+ (3)+ + + + 2 ⎩ γ u + γ u + γ u ,a ≤ 0. 1 1 2 1 2 1 j+ j+ j+ 2 2 2 a≥0 ⎪ (1)− 2−ν 1−ν u = u + + , ⎪ 1 ⎪ 1 j−1 3 2 j− ⎪ ⎧ j+ ⎪ − 1 γ = (1 + v)(2 + v), ⎪ ⎪ 1 20 ⎨ ⎨ (2)− 2 2−ν 1−ν − u = u + + , j 1 γ = (3 − v)(2 + v), 1 j 3 2 2 j− ⎪ j+ ⎪ ⎪ ⎩ ⎪ − γ = (3 − v)(2 − v), (3)− ⎪ 2 −1−ν 1−ν u = u + + , ⎩ j 1 1 j+1 3 2 j+ j+ a≤0 (1)+ 2 1−ν −1−ν u = u + + , ⎪ 1 j+1 ⎪ 1 j 3 2 j+ ⎪ ⎧ j+ ⎪ 2 + 1 γ = (2 + v)(3 + v), ⎪ ⎪ 1 20 ⎨ ⎨ (2)+ −2−ν −1−ν 2 + 1 u = u + + , j+1 3 γ = (2 − v)(3 + v), (11) 1 j+1 3 2 2 j+ ⎪ j+ ⎪ ⎪ 2 ⎩ ⎪ + γ = (2 − v)(1 − v). (3)+ ⎪ 2 −2−ν −1−ν u = u + + , ⎩ j+1 3 1 j+2 3 2 j+ j+ k m Final interpolation of Full-WENO5 can be written as Where ν = aτ/h, a = λ , is the m-th order differ- n n 2 3 ence, = u − u , = − , = 1 1 1 j+1 j j 1 j+ j+ j− j+ 2 2 2 2 − 2 2 ⎪ 1 − , etc. For nonlinear cases, we need to con- ⎪ 2 j +1 j − (ε+β ) (1)− sider the accuracy of λ,and we achieveitthrough theflux − − − ⎪ 1 ⎪ γ γ γ 1 2 3 j+ + + 2 reconstruction technique in section 3.2. The expression 2 2 2 ⎪ − − − ⎪ (ε+β ) (ε+β ) (ε+β ) ⎪ 1 2 3 of u ¯ in Equation (11) is deduced via Newton interpo- 1 ⎪ ⎪ − j+ ⎪ γ ⎪ 2 lation of υ (x), see Appendix A for detailed process. ⎪ (ε+β ) (2)− ⎪ + u − − − ⎪ 1 Smoothness indicatorfor Full-WENO5,from(Jiang& γ γ γ 1 2 3 j+ + + 2 ⎪ 2 2 2 − − − Shu, 1996) (ε+β ) (ε+β ) (ε+β ) 1 2 3 a≥0 3 ⎪ 2 (ε+β ) ⎪ (3)− − ⎪ 13 2 + u ,a ≥ 0, ⎪ − − − ⎪ β = (u − 2u + u ) 1 j−2 j−1 j ⎪ γ γ γ ⎪ 1 12 j+ ⎪ 1 2 3 ⎪ + + 2 2 2 2 ⎪ 1 2 − − − (ε+β ) (ε+β ) (ε+β ) ⎪ + (u − 4u + 3u ) , j−2 j−1 j 1 2 3 u ¯ = j+ ⎨ − 13 γ 2 2 ⎪ 1 β = (u − 2u + u ) j−1 j j+1 12 ⎪ 2 ⎪ (ε+β ) (1)+ ⎪ 1 ⎪ u + + + + (u − u ) , ⎪ 1 ⎪ j−1 j+1 4 ⎪ γ γ 1 2 3 j+ ⎪ + + 2 ⎪ 2 2 2 + + + ⎪ − 13 2 (ε+β ) (ε+β ) (ε+β ) 1 2 3 β = (u − 2u + u ) ⎪ ⎪ j j+1 j+2 3 12 ⎪ ⎪ + 1 2 γ + (3u − 4u + u ) , ⎪ j j+1 j+2 4 2 ⎪ (ε+β ) (2)+ ⎪ + u a≤0 + + + γ γ γ ⎪ 1 2 3 j+ + + 2 ⎪ 2 2 2 + + + 13 2 ⎪ (ε+β ) (ε+β ) (ε+β ) β = (u − 2u + u ) 1 2 3 j−1 j j+1 1 ⎪ ⎪ 12 1 2 ⎪ + (u − 4u + 3u ) , 3 j−1 j j+1 ⎪ ⎪ 4 ⎪ 2 ⎨ + (ε+β ) (3)+ + 13 1 ⎪ 2 2 + u ,a ≤ 0. ⎪ + + + β = (u − 2u + u ) + (u − u ) , 1 j j+1 j+2 j j+2 ⎪ γ γ γ 12 4 j+ ⎪ 1 2 3 ⎩ + + 2 ⎪ 2 2 2 + + + + 13 (ε+β ) (ε+β ) (ε+β ) ⎪ 1 2 3 ⎪ β = (u − 2u + u ) j+1 j+2 j+3 ⎪ 3 12 (13) + (3u − 4u + u ) . j+1 j+2 j+3 (12) ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 7 Where is a positive real number introduced to avoid the 3.1.2. Initial value reconstruction for seventh order −6 denominator becoming zero, and we will take = 10 full-WENO (Full-WENO7) in later tests. Equations (12) and (13) come from refer- The fully discrete interpolation and weighted coefficients ence (Jiang & Shu, 1996). of Full-WENO7 can be given as. 2 3 4 5 6 4−ν −2−ν 3−ν −1−ν 2−ν 1−ν u + ( + ( + ( + ( + ( + ) ) ) ) ) , a ≥ 0, ⎪ j j 1 j 1 j 7 6 5 4 3 2 j− j− j− 2 2 u ¯ = j+ ⎪ 2 3 4 5 6 −4−ν 2−ν −3−ν 1−ν −2−ν −1−ν u + ( + ( + ( + ( + ( + ) ) ) ) ) , a ≤ 0, ⎩ j+1 j+1 3 j+1 3 j+1 7 6 5 4 3 2 j+ j+ j+ 2 2 weighted by 4 fourth order stencils −→ ←− finial seventh order stencils ⎪ − (k)− γ u ,a ≥ 0, j+ k=1 u ¯ = j+ (k)+ 2 + γ u ,a ≤ 0. k 1 j+ k=1 a≥0 (1)− 3−ν 2−ν 1−ν n 2 3 u = u + ( + ( + ) ) , ⎪ 1 1 j j−1 3 4 3 2 j− ⎪ j+ j− ⎪ 2 2 (2)− ⎪ −1−ν 2−ν 1−ν n 2 3 u = u + ( + ( + ) ) , ⎨ 1 1 j j 1 4 3 2 j− j+ j− 2 2 (3)− 3 −1−ν 2−ν 1−ν n 2 u = u + ( + ( + ) ) , ⎪ 1 1 j j 1 4 3 2 j− ⎪ j+ j+ ⎪ 2 2 (4)− ⎪ n 3 −2−ν −1−ν 1−ν u = u + ( + ( + ) ) , ⎩ 1 1 j j+1 3 4 3 2 j+ j+ j+ 2 2 − 1 ⎪ γ = (1 + v)(2 + v)(3 + v), 1 210 − 1 γ = (4 − v)(2 + v)(3 + v), ⎪ − 1 γ = (4 − v)(3 − v)(3 + v), 3 70 − 1 γ = (4 − v)(3 − v)(2 − v), a≤0 (1)+ n 2 3 2−ν 1−ν −1−ν u = u + ( + ( + ) ) , ⎪ 1 1 j+1 j 1 ⎪ 4 3 2 j+ ⎪ j+ j− 2 2 ⎪ (2)+ n 2 3 1−ν −2−ν −1−ν u = u + ( + ( + ) ) , ⎪ 3 1 j+1 j+1 1 ⎨ 4 3 2 j+ j+ j+ 2 2 ⎪ (3)+ n 2 3 1−ν −2−ν −1−ν u = u + ( + ( + ) ) , 1 j+1 j+1 3 4 3 2 ⎪ j+ j+ j+ ⎪ 2 2 (14) (4)+ −3−ν −2−ν −1−ν ⎪ n 2 3 u = u + ( + ( + ) ) , 1 j+1 j+2 5 4 3 2 j+ j+ j+ ⎧ 2 2 γ = (2 + v)(3 + v)(4 + v), 1 210 ⎨ + 1 γ = (3 − v)(3 + v)(4 + v), γ = (3 − v)(2 − v)(4 + v), ⎩ + 1 γ = (3 − v)(2 − v)(1 − v). 4 210 The expression of u ¯ in Equation (14) is deduced via j+ Newton interpolation of υ (x), the detailed process is the same as fifth order case and is omitted here. 8 H. DONG ET AL. Smoothness indicator for Full-WENO7, from (Balsara designed scheme is fully high order for linear equations. &Shu, 2000) Now we show the fully high-order version in nonlinear cases. If we want high-order nonlinear accuracy, we need a≥0 n + 1 to linearize the u fl x with exact solution at time t for β = u (547u − 3882u − 4642u ⎪ 1 j−3 j−3 j−2 j−1 rarefaction waves and compression waves before shocks −1854u ) + u (7043u − 1724u + 7042u ) ⎪ j j−2 j−2 j−1 j formed.However,wecannotgetthenonlinearexactsolu- +u (11003u − 9402u ) + 2107u , j−1 j−1 j j tion, so we need to reconstruct the quasi-exact solution n+1 n ⎪ β = u (267u − 1642u + 1602u − 494u ) 2 j−2 j−2 j−1 j j+1 u by thedataattime t to replace the arithmetic ⎪ j+ +u (2843u − 5966u + 1922u ) 2 j−1 j−1 j j+1 ⎨ 2 means in Equations (9) and (10). More specifically, the +u (3443u − 2522u ) + 547u , j j j+1 j+1 n+1 quasi-exact solution u origins from solution formula β = u (547u − 2522u + 1922u − 494u ) 3 j−1 j−1 j j+1 j+2 j+ ⎪ +u (3443u − 5966u + 1602u ) method for conserved variables, which actually is the j j j+1 j+2 ⎪ n +1 ⎪ +u (2843u − 1642u ) + 267u , first-order derivative of v in Equation (7): j+1 j+1 j+2 ⎪ j+2 j+ β = u (2107u − 9402u + 7042u − 1854u ) ⎪ 4 j j j+1 j+2 j+3 +u (11003u − 17246u + 4642u ) j+1 j+1 j+2 j+3 n+1 1 1 n 1 2 2 n 2 u = R (L v (x − λ τ)) + R (L v (x − λ τ)) +u (7043u − 3882u ) + 547u , 1 1 1 x x j+2 j+2 j+3 j+3 j+ j+ j+ 2 2 a≤0 3 3 n 3 + R (L v (x − λ τ)). (17) j+ β = u (2107u − 9402u + 7042u ⎪ 1 j+1 j+1 j j−1 −1854u ) + u (11003u − 17246u ⎪ j−2 j j j−1 k k k n+1 ⎪ Here the initial L , R , λ in u can be acquired by +4642u ) + u (7043u − 3882u ) ⎪ j−2 j−1 j−1 j−2 j+ 2 2 +547u , j−2 baseline u-entropy conditioninEquation(9) ⎪ β = u (547u − 2522u + 1922u k k k 2 j+2 j+2 j+1 j Then the reconstructed L , R , λ can be given by −494u )+u (3443u −5966u + 1602u ) ∗k j−1 j+1 j+1 j j−1 u-entropy condition and ϕ can be given p entropy ⎨ 2 +u (2843u − 1642u ) + 267u , j j j−1 condition. j−1 β = u (267u − 1642u + 1602u 3 j+3 j+3 j+2 j+1 ⎪ n n n n ⎪ −494u ) + u (2843u − 5966u + 1922u ) ρ u + ρ u j j+2 j+2 j+1 j j j j+1 j+1 ⎪ ⎪ n n ⎪ ⎪ u > u ⎪ ⎨ j j+1 n n ⎪ +u (3443u − 2522u ) + 547u , j+1 j+1 j ρ + ρ ⎪ j j j+1 u ≡ , β = u (547u − 3882u − 4642u ⎪ ⎪ 4 j+4 j+4 j+3 j+2 ⎪ n+1 n n ⎪ ⎩ u u ≤ u 1 j j+1 −1854u ) + u (7043u ⎪ j+ j+1 j+3 j+3 ⎪ ⎧ −1724u + 7042u ) j+2 j+1 n n n n ρ H + ρ H j j j+1 j+1 2 ⎪ n n +u (11003u − 9402u ) + 2107u . u > u j+2 j+2 j+1 ⎪ j+1 n n j j+1 ρ + ρ j j+1 (15) H ≡ , n+1 n n H u u ≤ u 1 j j+1 Final interpolation of Full-WENO7 can be written as j+ A ≡ A(u, c), ⎪ k ⎪ 4 − 2 L ≡ L(u, c), ⎪ (ε+β ) (k)− ⎪ k c = (γ − 1)(H − ) → u ,a ≥ 0, ⎪ − 2 4 ⎪ R ≡ R(u, c), ⎪ γ j+ ⎪ ⎪ k k=1 ⎩ ⎨ − λ ≡ λ(u, c), (ε+β ) k=1 u ¯ = (16) 2γ + ⎛ ⎛ ⎞ ⎞ j+ ⎪ γ ⎪ k γ −1 n n n n u −u c c ⎪ 2 4 + j j+1 j j+1 (ε+β ) ⎜ ⎜ + + ⎟ ⎟ ⎪ (k)+ 2 γ −1 γ −1 u ,a ≤ 0. ⎜ ⎜ ⎟ ⎟ ⎪ + p = max 0, , ⎪ γ γ −1 γ −1 ⎝ ⎝ n n ⎠ ⎠ j+ ⎪ k c c k=1 ⎩ 2 j j+1 1 1 2γ 2γ ( ) + ( ) n n (ε+β ) k=1 γ −1 p γ −1 p j j+1 ⎛ ⎞ n n n n u +u f(u )+f(u ) j j+1 j j+1 Equations (15) and (16)come from reference (Balsara & 1 n λ − p > p 2 2 j ⎜ ⎟ Shu, 2000). ∗1 1 ⎜ ⎟ ϕ ≡ L , ⎝ ⎠ n+1 n+1 1 n λ u − f u p ≤ p 1 1 j j+ j+ 2 2 3.2. Flux reconstruction for full-WENO ∗2 2 2 n+1 n+1 ϕ ≡ L λ u − f u , 1 1 This processistoreach designed accuracy fornonlin- j+ j+ 2 2 ⎛ ⎞ n n n n ear equations, to be exact, we need to obtain the local u +u f(u )+f(u ) j j+1 j j+1 3 n λ − p > p k k k ∗k j+1 2 2 constants L , R , λ , ϕ for the composition of nal fi ⎜ ⎟ ∗3 3 ⎜ ⎟ ϕ ≡ L . (18) ⎝ ⎠ numericalflux.Wehavegiven thebaselineentropy con- n+1 n+1 3 n λ u − f u p ≤ p 1 1 j+1 dition linearization in section 2.2 and guarantee that the j+ j+ 2 2 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 9 k k k Due to the accuracy order of L , R , λ getting from in section 3.1 to maintain the non-oscillation property. baseline u-entropy conditionisnot equaltothe recon- Details are shown as follows. n+1 structed u ,itwillinufl encethe accuracy to some j+ 3.2.1. Flux reconstruction for r = 3, full-WENO5 content. So, theoretically, iteration is needed for getting a n+1 2 2 2 2 ∗ ∗ more accurate u .However,wedonotrecommendthis if (β = min(β , β , β )) ⇒ u = u , j+ k 1 2 3 k 2 j+ ⎧ 2 ⎡ ⎤ process owing to the complexity of Euler equations and −(−v)(1−v) u + j 1 dv 2! ⎪ j− consideration of computing burden. Actually, the numer- ⎪ ⎢ ⎥ ⎪ ⎢ ⎥ −(−v)(1−v)(2−v) ⎪ 2 d ical experiments tell that the error gently corrected by ⎪ ⎢ ⎥ j−1 dv 3! ⎪ ⎢ ⎥ iteration contributesfew inufl encesinthe vast major- ⎪ ⎢ ⎥ d −(−v)(1−v) ⎪ ⎢ ⎥ u + j 1 dv 2! ity of tests, except for testing nonlinear accuracy order. ⎪ ⎢ j− ⎥ ⎪ ⎢ ⎥ , a ≥ 0, −(−v)(1−v)(2−v) Moreover, the greatest influence factors on resolutions ⎪ ⎢ 2 d ⎥ ⎪ ⎢ ⎥ dv 3! k ∗k ⎪ ⎪ ⎢ ⎥ are the reconstructed λ ϕ , so we even do not need to d −(−v)(1−v) ⎪ ⎢ ⎥ k k ⎪ u + j 1 ⎪ ⎢ ⎥ reconstruct L , R for almost all tests. It means we can dv 2! ⎪ j+ ⎪ 2 ⎡ ⎤ ⎣ ⎦ k k directly use the L , R in Equation (17) which is calcu- u d −(−1−v)(−v)(1−v) ⎨ 2 j+1 dv 3! ⎣ ⎦ ⎡ ⎤ lated by baseline entropy condition. By the way, we also u ≡ 2 d −(−1−v)(−v) u + ∗ j+1 1 dv 2! keep in line with the above statement in following numer- u ⎪ j+ 3 ⎪ ⎢ ⎥ ⎪ ⎢ ⎥ ical experiments. (Note: we can get the entropy condition ⎪ −(−1−v)(−v)(1−v) 2 d ⎪ ⎢ ⎥ dv 3! ⎪ ⎢ ⎥ linearization formula for the scalar equation if we remove ⎪ ⎪ ⎢ ⎥ ⎪ d −(−1−v)(−v) ⎪ ⎢ ⎥ u + j+1 the eigenvectors LR and use the eigenvalue for determin- ⎪ dv 2! ⎪ ⎢ j+ ⎥ ⎪ 2 ⎪ ⎢ ⎥ , a ≤ 0. ing the selection between Roe means and reconstruction, ⎪ ⎢ d −(−2−v)(−1−v)(−v)⎥ ⎪ ⎢ ⎥ j+1 dv 3! see details in Zhou & Dong, 2021, 2022) ⎪ ⎢ ⎥ −(−1−v)(−v) n+1 ⎪ ⎢ d ⎥ k ∗ u + Denote L u ≡ u ,accordingtothe accuracy 3 j+1 ⎪ ⎢ ⎥ 1 1 dv 2! j+ j+ j+ ⎪ ⎣ 2 ⎦ 2 2 ⎪ ⎩ d −(−2−v)(−1−v)(−v) order analysis in (Zhou & Dong, 2021, 2022), final accu- j+2 dv 3! racy order of scheme via solution formula method = min (19) (accuracy order of initial value reconstruction, 2 × accuracy order of ux fl reconstruction). So, we can obtain The expression of u in Equation (19) is deduced j+ designed order by r-th interpolation of u∗ + .While j 1 via Newton interpolation of υ (x), see Appendix A for doing r-th Newton interpolation for initial value υ (x)in detailed process. The expansion form of Equation (19) is section 3.1, we can obtain their corresponding rfi st-order in Appendix A derivative function u (x). Inserting x = x -νh,weget j+ 3.2.2. Flux reconstruction for r = 4, full-WENO7 the n fi al formula. And we use the smoothness indicators 2 2 2 2 2 ∗ ∗ if (β = min(β , β , β , β )) ⇒ u = u , k 1 2 3 4 k j+ ⎡ ⎤ d −(−v)(1−v) d −(−v)(1−v)(2−v) d −(−v)(1−v)(2−v)(3−v) 2 3 u + + + j 1 dv 2! j−1 dv 3! 3 dv 4! ⎪ j− j− ⎪ 2 ⎢ ⎥ ⎢ −(−v)(1−v) −(−v)(1−v)(2−v) −(−1−v)(−v)(1−v)(2−v) ⎥ d 2 d 3 d ⎪ u + + + ⎢ j ⎥ ⎪ j 1 dv 2! dv 3! dv 4! j− j− ⎢ ⎥ ⎢ ⎥ , a ≥ 0, d −(−v)(1−v) 2 d −(−v)(1−v)(2−v) 3 d −(−1−v)(−v)(1−v)(2−v) ⎢ ⎥ u + + + ⎪ 1 j 1 ⎪ dv 2! dv 3! dv 4! ⎢ j− ⎥ ⎡ ⎤ j+ ⎪ 2 ⎣ ⎦ ⎪ d −(−v)(1−v) d −(−1−v)(−v)(1−v) d −(−2−v)(−1−v)(−v)(1−v) 2 3 u + + + ⎨ j 1 ∗ dv 2! j+1 dv 3! 3 dv 4! ⎢ ⎥ j+ u j+ ⎢ ⎥ 2 ≡ ⎡ ⎤ ∗ −(−1−v)(−v) −(−1−v)(−v)(1−v) −(−1−v)(−v)(1−v)(2−v) ⎣ ⎦ d 2 d 3 d u ⎪ u + + + j+1 1 3 ⎪ j 1 dv 2! dv 3! dv 4! ⎪ j+ j− ⎪ 2 ⎢ ⎥ u ⎪ 4 ⎪ −(−1−v)(−v) −(−2−v)(−1−v)(−v) −(−2−v)(−1−v)(−v)(1−v) ⎢ d 2 d 3 d ⎥ ⎪ u + + + ⎢ j+1 ⎥ ⎪ j+1 1 dv 2! dv 3! dv 4! ⎪ j+ ⎢ j+ ⎥ ⎢ ⎥ , a ≤ 0. d −(−1−v)(−v) d −(−2−v)(−1−v)(−v) d −(−2−v)(−1−v)(−v)(1−v) 2 3 ⎢ ⎥ ⎪ u + + + j+1 3 ⎪ dv 2! j+1 dv 3! 3 dv 4! ⎢ j+ ⎥ ⎪ j+ ⎣ ⎦ ⎪ −(−1−v)(−v) −(−2−v)(−1−v)(−v) −(−3−v)(−2−v)(−1−v)(−v) d 2 d 3 d u + + + ⎩ j+1 3 j+2 5 dv 2! dv 3! dv 4! j+ j+ (20) 10 H. DONG ET AL. Where the smoothness indicators β in u fl x applied in initial value reconstruction for saving com- reconstruction can also be calculated by Equation (12) puting time, which means the β applied in initial value (for Full-WENO5) and Equation (15) (for Full-WENO7). reconstruction and ux fl reconstruction are same. The With the minimum squared value of β ,wecan getthe expansion form of Equation (20) is in Appendix A. smoothest stencil for ux fl reconstruction and avoid the Additionally, for some cases with high demand in suspicious oscillations at risk regions as much as possi- robustness, such as Test 12 Double Mach reflection in ble. Moreover, the β we get in this step can be directly section 5.4, u fl x reconstruction requires an order reduc- tion.We’dliketodoasfollows. 2 2 2 2 u , if < 0 ∩ < 0 ∩ < 0 , a ≥ 0, ⎪ 1 1 j−1 j j j+1 j− j+ 2 2 u = (21) j+ ⎪ 2 ⎪ 2 2 2 2 u , if < 0 ∩ < 0 ∩ < 0 , a ≤ 0. j+1 1 3 j j+1 j+1 j+2 j+ j+ 2 2 Equation (21) means we do the order reduction (Strang, 1968) and turn it into three one-dimensional whilemiddleconvexity is dieff rent with twoendsof equations. it (nonconvex-convex-nonconvex, convex-nonconvex- ⎛ ⎞ convex). This method balances robustness and resolution ⎜ ⎟ ρu to some extent. ⎜ ⎟ ⎜ ⎟ u + f (u) = 0 ⇔ ρv Note 1: The entropy condition of solution formula e ⎜ ⎟ ⎝ ⎠ e=ξ,η,ζ ρw method and smoothness indicator of WENO guarantee high resolution and non-oscillation properties together. ⎛ ⎞ Then Full-WENO can be composed as follows. ρ(e u + e v + e w) x y z ⎜ ⎟ Full-WENO5 is composed of Equations (7–13) and (18 ρ(e u + e v + e w)u + e p(u) x y z x ⎜ ⎟ ⎜ ⎟ and 19). + ρ(e u + e v + e w)v + e p(u) = 0, x y z y ⎜ ⎟ ⎝ ⎠ e=ξ,η,ζ ρ(e u + e v + e w)w + e p(u) Full-WENO7 is composed of Equations (7–10) and x y z z (14–16) and (18 and 20). (e u + e v + e w)(E + p(u)) x y z Note 2: If we remove the u fl x reconstruction, take ν = 0 ⇓ dimensional splitting and associate it with RK method, the scheme in this paper ⎛ ⎞ will degrade to a normal semi-discrete WENO + RK ⎜ ⎟ ρu ⎜ ⎟ scheme. ⎜ ⎟ u + f (u) = 0 ⇔ ρv t e ⎜ ⎟ Note 3: If we remove the eigenvectors, solution for- ⎝ ⎠ ρw mula, and entropy condition, then use RK method to tracethe characteristic line,wecan getthe SL/EL-WENO ⎛ ⎞ (Huang et al., 2016) for scalar nonlinear equation. What ρ(e u + e v + e w) x y z ⎜ ⎟ we mean is Full-WENO may quite different from SL/EL- ρ(e u + e v + e w)u + e p(u) x y z x ⎜ ⎟ ⎜ ⎟ WENO (Huang et al., 2016; Huang & Arbogast, 2017), + ρ(e u + e v + e w)v + e p(u) = 0, x y z y ⎜ ⎟ ⎝ ⎠ especially for systems. ρ(e u + e v + e w)w + e p(u) x y z z Note 4: In theory, with same computational condition, (e u + e v + e w)(E + p(u)) x y z the computing speed of Full-WENO will be s times faster (e = ξ, η, ζ). (22) than same order WENO with s-stage RK method. For example, for six-stage RK5 and nine-stage RK7 (Butcher, Equation (1) and the last equation of Equation (22) is 2016) (see Appendix B), the computing speed of Full- absolutely thesameformexceptthatthe number of com- WENO5willbe6timesfasterthanWENO5 + RK5, and ponents is different. So it is easy to generalize all for- Full-WENO7 will be 9 faster than WENO7 + RK7. mulas of schemes (Full-WENO5 and Full-WENO7) for Equation (22) from Equation (1). For example, the gen- eral expression of numerical u fl x for conservative scheme 4. Multi-dimensional cases in 3D can be written as follows For three-dimensional Euler equations in curvilin- e e n+1 n τ ˜ ˜ u = u − (f 1 − f 1 ), e = ξ, η, ζ , j h ear coordinates, we use dimensional splitting method j+ j - 2 2 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 11 Figure 1. Accuracy order test, linear equation. (A) 5-th order schemes (B) 7-th order schemes. 5.1.1. Test 1 Accuracy order test, linear equation k = 1, 2, 3, 4, 5, This is an accuracy order test of linear equation, initial e,k e,k e,k ∗e,k value is given as follows. f 1 = R (λ L u ¯ − ϕ ), j+ j+ k=1 u + u = 0, t = 2, t x n n e,k (24) v −v (e−λ τ) u (x) = sin(πx), x ∈ [−1, 1]. j+ 0 e,k e,k L u ¯ ≡ L . (23) e,k λ τ j+ 5.1.2. Test 2 Accuracy order test, Burgers equation This is an accuracy order test of nonlinear equation, initial value is given as follows. More detailed information on three-dimensional cases can be found in our former work (Zhou & Dong, 2021, 2 u + = 0, t = 0.15, 2022). (25) u (x) = 0.5 + sin(πx), x ∈ [0, 2]. 5. Numerical experiments As shown in Figures 1 and 2,bothFull-WENO and WENO + RK almost achieve designed accuracy order. This section we do numerical experiments for Full- Full-WENO reaches similar accuracy order compared WENO, which contains accuracy order tests, scalar with the same order WENO +RK,however,withsmaller equation tests, Euler equation tests, and sonic point test. errors. While WENO + RK3 only shows about third For all tests, we compare our new schemes with fre- order under CFL = 0.5 duetothe spacediscretedoesnot quently used WENO-RK schemes. If not specifically dominate the accuracy order at this CFL number. In test 1 stated WENO-RK uses TVD-RK3 (Jiang & Shu, 1996) Full-WENO7 and WENO7 + RK7 only reach about sixth (see Appendix B) and Local Lax-Friedrichs (LLF) ux fl orderand they will reachabout eighth orderifwecon- (Svenn Tveit, 2011). All numerical simulations are cal- tinue to ren fi e the grids. After further research, we found culatedbyour owncodes developedinFortran95 with this phenomenon may cause by the positive real number Microsoft Visual Studio 2013 and no specific data is used. in smoothness indicator, which has been discussed in reference (Henrick et al., 2005). 5.1. Tests of accuracy order 5.1.3. Test 3 Point-by-point accuracy order (PbP This subsection we test accuracy order of new scheme, order) which contains linear equation, Burgers equation, and Here we construct a new method for testing accuracy point-by-point accuracy order tests. order. As the name suggests, point-by-point accuracy 12 H. DONG ET AL. Figure 2. Accuracy order test, Burgers equation. (A) 5-th order schemes (B) 7-th order schemes. Table 1. Accuracy order test, theoretical error of schemes. Theoretical error R (a = 1, T = 2, v = CFL) CFL = 0.1 CFL = 0.5 CFL = 0.9 CFL = 1.0 2 2 2 (1−ν )(2 −ν )(3−ν) 5 5 5 5 5 Full-WENO5 R = (aT| |2π )h 19.475 h 11.954 h 2.164 h 0 Full - WENO5 6! 2 2 2 2 2 (1−ν )(2 −ν )(3 −ν )(4−ν) 7 7 7 7 7 Full-WENO7 R = (aT| |2π )h 41.497 h 25.808 h 4.611 h 0 Full - WENO7 8! 2 ×3 5 5 5 WENO5 R = (aT| |2π )h 20.401 h WENO5 6! 2 2 2 ×3 ×4 7 7 7 WENO7 R = (aT| |2π )h 43.147 h WENO7 8! orderistoreckonthe accuracy orderatevery point without the consideration of the error origin from RK by applying the theoretical error and accuracy. This method. When CFL→1 the theoretical error of Full- method can visually locate positions of order loss and is WENO tends to be zero, and when CFL→0ittends to helpful for researchers checking and analyzing bugs of be same as semi-discrete WENO (for nonlinear cases algorithms. It’s meaningful for research of smoothness Full-WENOneedstoconsiderthe errorof ν = λτ/h). indicators. Consider following linear equation. Figure 3 shows the point-by-point accuracy order, where optimal value is calculated directly by optimal weight u + au = 0, t ≤ T, t x without smoothness indicator. Figure 3 also tells that (26) a = 1, T = 2. Full-WENO achieves similar accuracy order compared with same order WENO + RK, however, WENO + RK3 point-by-point accuracy order can be given as only achieves third to fourth order. And we can also nd, fi it may introduce new errors after applying smoothness R(x) q(x) = ln / ln h. (27) C(x) indicator and thus trigger an overall accuracy reduction, which means there is still room for improvement in this (q+1) |u (x)| (q+1) Where C(x) = Cu = C , and theoretical smoothness indicator. The accuracy order test method in (q+1) |u | error R = Ch has been given in Table 1, C is theo- Test 1 may mask this problem due to the division of two retical error coefficient, q is theoretical accuracy order, errors at different grids, and nall fi y get designed accuracy. (q + 1) (q+1) |u (x)| is q-th derivative of initial value, |u | is theaverage valueof q-th derivative, h is space step. Here we test an inn fi itely smooth case, initial value can 5.2. Scalar equation be foundinEquation(24),and forthiscasewehave 5.2.1. Test 4 Linear scalar equation with multiple q+1 π cos(πx) , q = 2k − 1, (q+1) 2π extremes u = k = 1, 2, 3 ... q+1 π sin(πx) This test is composited by a series of smooth and , q = 2k, 2π Table 1 shows that, for linear cases, the theoretical unsmooth functions, which contains a Gaussian, a error of WENO + RK is not relevant to CFL number square, a triangle, and a semi-ellipse. It is widely used ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 13 Figure 3. Accuracy order test, point-by-point accuracy. (A) 5-th order schemes (B) 7-th order schemes. (G(x, β, z − δ) + G(x, β, z + δ) + 4G(x, β, z)), −0.8 ≤ x ≤−0.6, ⎪ 6 1, −0.4 ≤ x ≤−0.2, u (x) = 1 −|10(x − 0.1)|,0 ≤ x ≤ 0.2, ⎪ (F(x, α, a − δ) + F(x, α, a + δ) + 4F(x, α, a)), 0.4 ≤ x ≤ 0.6, (28) 0, else, −β(x−z) 2 G = e , F(x, α, a) = max(1 − α (x − a) ,0), z =−0.7, δ = 0.005, β = (log 2)/(36δ ), a = 0.5, α = 10. The result of Figure 4 is solved with 200 points and tells that Full-WENO is also equipped with an excel- t = 8, which shows the quite different solution properties lent low dissipative property in nonlinear cases, which is between these two schemes. Combining Table 1,wecan mainly attributed to the consistent high-order spatial and know theresolutionofFull-WENO forthislinearcase temporal accuracy. tend to be similar with semi-discrete WENO + RK when CFL→0and to be exactsolutionwhenCFL→1. On the 5.3. One-dimensional Euler equations contrary, due to RK3 method needs a small CFL number to maintain its robustness and accuracy, the resolution of 5.3.1. Test 6 Sod shock tube problem This is a typical Riemann problem for 1D Euler equa- WENO + RK3 performs worse with the increasing CFL number tions. Initial value is given as follows. (1, 0, 1),0 ≤ x ≤ 0.5, (ρ, u, p) = (29) (0.125, 0, 0.1), 0.5 ≤ x ≤ 1. 5.2.2. Test 5 Burgers equation This is a nonlinear case, the initial value has been given The solutions at t = 0.2 with 200 points, CFL = 1for in Equation (25). Forty points and period boundary Full-WENO, CFL = 0.4 for WENO + RK3 and recon- are used, and we output the solution at t = 0.5 and structed eigenvectors are plotted in Figure 6.Itcan be t = 20 respectively. And we set CFL = 1forFull-WENO, seen that, owing to the high order u fl x reconstruction CFL = 0.4 for WENO + RK3. Figure 5 shows that Full- for rarefaction waves and linear field, Full-WENO resolve WENO performs better at the discontinuity and still the rarefaction waves and contract discontinuity much keeps a high resolution after a long-time evolution, while better than WENO + RK3. Also, owing to Roe means are WENO + RK3 obviously loses its resolution. This test applied for the compression waves that may evolve into 14 H. DONG ET AL. Figure 4. Linear scalar equation, multiple extremes case. (A) Full-WENO5, overall view (B) Full-WENO5, local view (C) WENO5 + RK3, overall view (D) WENO5 + RK3, local view (E) Full-WENO7, overall view (F) Full-WENO7, local view (G) WENO7 + RK3, overall view (H) WENO7 + RK3, local view. ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 15 Figure 5. Nonlinear scalar equation, Burgers equation. (A) 5-th order schemes, t = 0.5 (B) 7-th order schemes, t = 0.5 (A) 5-th order schemes, t = 20 (B) 7-th order schemes, t = 20. shocks, the performance of Full-WENO at the shock is 10000 points. It is shown in Figure 7 that Full-WENO also better than WENO + RK3 with LLF u fl x. resolves the density profile much better than same order WENO + RK3, especially near the valley x = 0.75 and 5.3.2. Test 7 Blast-wave problem the right peak x = 0.78, which means that Full-WENO We now consider the interaction of two blast waves. exhibits less dissipation than WENO + RK3. Initial value is given as follows. 5.3.3. Test 8 Shu-Osher problem (1, 0, 10 ),0 ≤ x < 0.1, This is a typical example for testing the performance −2 (ρ, u, p) = (1, 0, 10 ), 0.1 ≤ x < 0.9, (30) ⎩ of high-order schemes when the solution contains both (1, 0, 10 ), 0.9 ≤ x < 1. shocks and complex smooth region structures. In this A reflective boundary condition is imposed at both case, a one-dimensional Mach 3 shock wave interacts ends. The simulation is performed with 500 points until with a perturbed density eld fi generating both small- final time t = 0.038. The reference solution is obtained scale structures and discontinuities, hence it is selected to by second-order TVD scheme (Harten, 1983)with validate shock-capturing and wave-resolution capability. 16 H. DONG ET AL. Figure 6. 1D Euler equations, Sod shock tube problem. (A) 5-th order schemes, overall view (B) 5-th order schemes, local view (C) 7-th order schemes, overall view (D) 7-th order schemes, local view. Initial value is given as follows. test well confirms that Full-WENO is less dissipation than WENO + RK3, which mainly attributes to its high-order (3.857, 2.629, 10.333), −5 ≤ x < −4, accuracy in time evolution. (ρ, u, p) = (1 + 0.2 sin(5x),0,1), −4 ≤ x < 5. (31) 5.4. Two-dimensional Euler equations Here we solve the case with 200 and 400 grids 5.4.1. Test 9 Two-dimensional Riemann problem respectively, and we set CFL = 1for Full-WENO, This is a Riemann problem with interaction of planar CFL = 0.4 for WENO + RK3. The reference solution is shocks. Initial value is given as follows. also obtained by second-order TVD scheme with 10,000 points. Figure 8 is the solution output at t = 1.8, which tells that Full-WENO performs much better than same order WENO +RK3bothatcoarseand nfi egrids.This (p, u, v, p) ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 17 Figure 7. 1D Euler equations, Blast-Wave problem. (A) 5-th order schemes (B) 7-th order schemes. Where indicates vortex strength, α controls the decay (0.138, 1.206, 1.206, 0.029), x < 0.8, y < 0.8, rate of vortex, r is the critical radius to control the max- (0.5323, 0, 1.206, 0.3), x > 0.8, y < 0.8, c imum vortex strength. Here we set = 0.3, r = 0.05, (1.5, 0, 0, 1.5), x > 0.8, y > 0.8, c α = 0.204. The computational domain is taken as (0.5323, 1.206, 0, 0.3), x < 0.8, y > 0.8. (x, y)∈[0, 2] × [0, 1], and the mesh of 200 × 80 is (32) used. We set CFL = 1for Full-WENO, CFL = 0.4 for We solvethe 2D Eulerequations in acomputational WENO + RK3. Figures 10 and 11 are the solution output domain of (x, y)∈[0,1] × [0,1] with the mesh resolu- at t = 0.6. tion 400 × 400 until nal fi time t = 0.8. We set CFL = 1 Figure 10 is the comparison of the density along for Full-WENO, CFL = 0.4 for WENO + RK3. Figure 9 thecenterlineof y = 0.5, which shows Full-WENO shows that Full-WENO resolves much better than same has less numerical dissipation according to the perfor- order WENO + RK3 in predicting the small-scale struc- mance of moving vortex. And Full-WENO also shows tures its excellent capturing capacity in Figures 10 and 11,for which the shock of Full-WENO is much sharper than 5.4.2. Test 10 Shock vortex interaction problem WENO + RK3. It is mainly because the Roe means inlaid This model problem describes the interaction between in u fl x linearization is friendly to shock. However, LLF a stationary shock and a vortex. At the initial moment, flux applied in WENO + RK3 may introduce some dissi- a stationary Mach 1.1 shock is positioned at x = 0.5 pation. Moreover, if use Roe flux for WENO + RK3, some and normal to the x-axis. Its left state is (p, u, v, p) = entropy xfi may be needed to maintain its robustness and (1, γ,0,1), and its right state can be obtained through correctness which may also introduce some dissipation, Hugoniot relation. A small vortex is superposed to the see analysis in section 5.6. flow left to the shock and centers at ( x , y ) = (0.25, 0.5). c c The state of vortex can be described as a perturbation 5.4.3. Test 11 Rayleigh-Taylor instability problem to velocity (u, v), temperature (T = p/ρ)and entropy This problem illustrates the flow caused by the instability (S = ln(p/ρ )) of the mean flow, then we can denote it in domain (x, y)∈ [0, 0.25] × [0, 1]. Initial value is given by the tilde values. as follows. α(1−τ ) u ˜ = ετe sin θ, α(1−τ ) v ˜ =−ετe cos θ, (ρ, u, v, p) 2 2α(1−τ ) (γ −1)ε e T =− , (33) 4αγ (2, 0, −0.025a · cos(8πx),2y + 1),0 ≤ y < 0.5, S = 0, (1, 0, −0.025a · cos(8πx), y + 1.5), 0.5 ≤ y ≤ 1. 2 2 (34) τ = r/r , r = (x − x ) + (y − y ) . c c c 18 H. DONG ET AL. Figure 8. 1D Euler equations, Shu-Osher problem. (A) 5-th order schemes, 200 points (B) 7-th order schemes, 200 points (C) 5-th order schemes, 400 points (D) 7-th order schemes, 400 points. andspace of Full-WENOwillleadtolessnumerical Where, a = γ p/ρ γ = is the sound speed. The dissipation. Moreover, a more relaxed CFL lead to fewer boundary conditions are set as follows: the initial con- time steps for Full-WENO, which also helps to reduce the dition at the bottom boundary is (ρ, u, v, p) = (2, 0, numerical dissipation. 0, 1); while at the top boundary (ρ, u, v, p) = (1, 0, 0, 2.5) is assigned; at the left and right boundaries, the reflecting boundary conditions are used. We solve the two-dimensional Euler equations by adding the source 5.4.4. Test 12 Double Mach reflection problem term (0, 0, ρ, ρv). Mesh 240 × 960 is used in this test, This is a classic test for investigating high-resolution and we set CFL = 1for Full-WENO, CFL = 0.4 for schemes. We solve the 2D Euler equations in a compu- WENO + RK3. We output the result at t = 1.95. As tational domain of (x, y)∈[0, 4] × [0, 1], and the mesh of described in Figure 12,Full-WENO hasresolvedmuch 1600 × 400 is adopted. Here we also set CFL = 1forFull- richer vortical structures than same order WENO. This WENO, CFL = 0.4 for WENO + RK3. At the bottom of is because the uniform high order accuracy in time computational domain, it is a reflecting wall. Initially, a ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 19 Figure 9. 2D Euler equations, density distribution of 2D Riemann problem, 30 contours from 0.12 to 1.76. (A) Full-WENO5 (B) WENO5 (C) Full-WENO7 (D) WENO7. strong shockofMach10makesa60° anglewiththe bot- 5.5. CPU time test tom wall which begins at x = 1/6 and extends to the top This section we test the CPU time (in sec.) of 1D and 2D of the domain at y = 0. Ahead of the shock, the initial tests respectively. All tests are obtained by same worksta- condition is ρ = 1.4, p = 1.0, u = 0, v = 0, and the exact tion (AMD Ryzen Threadripper 3790X 32-Core Proces- post-shock conditionisimposed by Hugoniot relation. sor, 3.69 GHz) and same working environment. As shown in Figure 13, Full-WENO shows its merits of high resolution and low dissipation by resolving much more small structures compared with WENO + RK3. 5.5.1. Test 13 CPU time test, one-dimensional Euler We also provide the original single entropy condition equations whichisguidedonlybyvelocity. In Figure 13,wecan Here we compare the efficiency of fully discrete Full- n fi d the resolution of double entropy condition improves WENO and semi-discrete WENO +RK.Herewechoose much compared to that of single entropy condition (the the Sod shock tube, initial value sees Equation (29). In reason has been explained in section 2.2). theory, the computing speed of Full-WENO should be 20 H. DONG ET AL. Figure 10. 2D Euler equations, shock vortex interaction, density at y = 0.5. (A) 5-th order schemes (B) 7-th order schemes. Table 2. CPU time test, Two-dimensional Riemann problem. s/CFL (CFL = (CFL of WENO + RK)/ semi/fully semi/fully (CFL of Full-WENO)) times faster than same order Two-dimensional Riemann problem, grid number = 400× 400, t = 0.8, 28 threads WENO with s-stage RK method. As described in Figure scheme CFL CPU time (Rate) CFL CPU time (Rate) 14, for 3-stage RK3, 6 stage-RK5, and 9-stage RK7, the r = 3 CPU time tests all basically meet this law. Taking RK3 Full-WENO5 0.4 74.7 (2.44) 1.0 30.6 (1.00) as an example: (1) With same CFL number, the com- WENO5 + RK3 0.4 206.1 (6.74) 1.0 84.0 (2.75) r = 4 puting speed of Full-WENO achieve about 2.8 times Full-WENO7 0.4 88.4 (2.44) 1.0 36.2 (1.00) compared with same order WENO + RK3; (2) Consid- WENO7 + RK3 0.4 244.1 (6.75) 1.0 100.8 (2.79) ering WENO + RK3 requires a small CFL number for its Table 3. CPU time test, Shock vortex interaction. resolution, with different CFL numbers, the computing speed of Full-WENO (CFL = 1) achieves about 6.8 times Shock vortex interaction, grid number = 200× 80, t = 0.6, 1 thread scheme CFL CPU time (Rate) CFL CPU time (Rate) compared with WENO + RK3 (CFL = 0.4). r = 3 More specicfi ally, here we extract a data from Figure 14 Full-WENO5 0.4 8.7 (2.23) 1.0 3.9 (1.00) (800 points) to conclude the computing speed compari- WENO5 + RK3 0.4 24.9 (6.38) 1.0 11.2 (2.87) r = 4 son in same order. Figure 15 tells that compared to the Full-WENO7 0.4 11.2 (2.15) 1.0 5.2 (1.00) same time-space accuracy WENO + RK, Full-WENO WENO7 + RK3 0.4 32.8 (6.31) 1.0 15.0 (2.92) is extremely equipped with competition in efficiency. The computing speed of Full-WENO5 can reach about 5.5 times as much as WENO5 + RK5 and Full-WENO7 WENO + RK3 under CFL = 1, and achieve about 6.8 canreach about8timesasmuchasWENO7 + RK7 times if WENO + RK3 uses CFL = 0.4 with considera- under same computing conditions, which also basically tion of robustness and resolution. Because the entropy conforms to the theoretical speed (namely 6 and 9 condition applies different strategies for compression times, respectively). Moreover, the storage memory of wavesand rarefactionwaves (lesscomputing burden for RK method is also more than s times compared to Full- compression waves), some differences may happen in WENO. CPU time rate. (Note: We do not consider the CPU time of source terms in Rayleigh-Taylor instability problem) 5.5.2. Test 14 CPU time test, two-dimensional Euler equations 5.6. Sonic point test The CPU time test of all 2D cases has been given in the corresponding test under section 5.4 (in sec). From For schemes do not satisfy the entropy condition may Tables 2–5, we can obviously n fi d the computing speed render anon-physicalsolutionatthe rarefactionwave of Full-WENO achieve about 2.8 times as much as with a sonic point, such as WENO with Roe ux fl (just ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 21 Figure 11. 2D Euler equations, density distribution of shock vortex interaction, 30 contours from 1 to 1.22. (A) Full-WENO5 (B) WENO5 (C) Full-WENO7 (D) WENO7. Table 4. CPU time test, Rayleigh-Taylor instability problem. Table 5. CPU time test, Double Mach reﬂection problem. Rayleigh-Taylor instability problem, grid number = 240× 960, Double Mach reﬂection problem, grid number = 1600× 400, t = 0.2, t = 1.95, 28 threads 28 threads scheme CFL CPU time (Rate) CFL CPU time (Rate) scheme CFL CPU time (Rate) CFL CPU time (Rate) r = 3 r = 3 Full-WENO5 0.4 279.1 (2.42) 1.0 115.1 (1.00) Full-WENO5 0.4 297.4 (2.42) 1.0 123.9 (1.00) WENO5 + RK3 0.4 792.1 (6.88) 1.0 330.2 (2.87) WENO5 + RK3 0.4 846.2 (7.07) 1.0 356.8 (2.88) r = 4 r = 4 Full-WENO7 0.4 335.1 (2.41) 1.0 139.1 (1.00) Full-WENO7 0.4 383.0 (2.40) 1.0 159.6 (1.00) WENO7 + RK3 0.4 935.0 (6.72) 1.0 388.6 (2.79) WENO7 + RK3 0.4 1090.1 (6.83) 1.0 448.5 (2.81) like that Roe scheme does not satisfy entropy condi- tion). However, Full-WENO can ecffi iently avoid the 5.6.1. Test 15 Sonic point test, reversed shock problem non-physical solution by formula solution method con- We design areversedshock problemthatcontainsasonic structed with entropy condition linearization. pointatrarefaction.Fivehundred points areused, initial 22 H. DONG ET AL. Figure 12. 2D Euler equations, density distribution of Rayleigh-Taylor instability problem, 50 contours from 0.9 to 2.2. (A) Full-WENO5 (B) WENO5 (C) Full-WENO7 (D) WENO7. value is given as follows. high accuracy order, high resolution to discontinuities, and high speed of computation, when compared to tra- (5.4, 7/9,31/3), −0.5 ≤ x ≤ 0, (ρ, u, p) = (35) ditional WENO-RK. The accuracy order tests verify that (1.4, 3.0, 1.0),0 ≤ x ≤ 1.5. new schemes reach the designed accuracy order. Scalar In this test, we output the result at t = 0.3 and use the tests including multiple extremes test and nonlinear fifth-order scheme as an example. WENO5 + RK3 with test show that new scheme’s resolution to discontinu- Roe ux fl is set as the comparison, and is imposed with ities and extremes is obviously higher than WENO- an entropy xfi Q(λ)(Harten, 1983)( = an extra articfi ial RK. One-dimensional Euler equation tests including Sod viscosity at the sonic point) shock tube, Blast-wave, and Shu-Osher, prove that the solution formula method is feasible in u fl id flow sim- |λ|, |λ| >ε, ulations, and keeps the priority to WENO-RK. Two- Q(λ) ≡ 0 ≤ ε ≤ 1. (36) 2 2 λ +ε ,|λ|≤ ε, dimensional Euler equation tests including 2D-Reimann, 2ε Shock-vortex, RT-instabilities, and Double Mach reflec- As shown in Figure 16,WENO5-Roe hasalargeunphys- tion show more cases of high-quality results of new ical discontinuity and tends to be normal only if an schemes using double entropy conditions, such as cap- extra articfi ial viscosity is used. While Full-WENO5 has turing much more n fi e structures than WENO + RK. well resolved the solution owing to its entropy condition CPU time test shows that new scheme has a speed of solution formula without any articfi ial viscosity. computation many times faster than WENO-RK. Sonic point test shows that new scheme can effectively avoid 5.7. Summary for numerical experiments the unphysical discontinuity without any extra arti-fi Overall, the numerical tests show that Full-WENO is cial viscosity, thanks to entropy condition inlaid in u fl x equipped with the merits of high ecffi iency including linearization. ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 23 Figure 13. 2D Euler equations, density distribution of Double Mach reﬂection problem, 30 contours from 2 to 22. (A) Full-WENO5 (double entropy condition) (B) Full-WENO5 (single entropy condition) (C) WENO5 (D) Full-WENO7 (double entropy condition) (E) Full-WENO7 (single entropy condition) (F) WENO7. 24 H. DONG ET AL. Figure 14. CPU time test, one-dimensional Euler equations, Sod shock tube problem. (A) 5-th order schemes (B) 7-th order schemes. Figure 15. CPU time test, computing speed comparison in same order. 6. Conclusions The solution formula method is a construction method Figure 16. Sonic point test, reversed shock problem. for difference schemes which we are in favor of. It is equipped with following merits: clear philosophy, easy derivation, natural upwind, and consistent space-time accuracy achievable.Inthispaper,weconstruct aFull- all semi-discrete schemes propelled with RK method do WENO scheme by solution formula method combined not have this merit, which indicates that Full-WENO with some techniques of WENO reconstruction. The can reserve more accurate information of solutions. new scheme has following vfi e highlights: (1) One- (3) High resolution: Full-WENO resolves better than step and fully discrete: Full-WENO gets rid of RK same order WENO + RK for all CFL numbers. More method and achieves consistent high accuracy order importantly, the larger the CFL number, the more obvi- both in space and time. And Full-WENO can per- ous the performance gap will be; (4) High computing form robustly under CFL = 1; (2) The excellent perfor- speed: The computing speed of Full-WENO can reach mance along with CFL number: When CFL→1, Full- about s timesasmuchassemi-discrete WENO + s- WENO not only tends to be exact in linear cases, but stage RK under same computational condition. Specif- also to be more accurate in nonlinear cases. This is ically, Full-WENO5 is nearly 5 ∼6times faster than named the traveling wave solution property. However, WENO5 + RK5, Full-WENO7 is nearly 8 ∼9timesfaster ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 25 than WENO7 + RK7; (5) Double entropy condition lin- Dumbser, M. (2014). Arbitrary-Lagrangian–Eulerian ADER– WENO finite volume schemes with time-accurate local earization: Full-WENO uses the newly designed double time stepping for hyperbolic conservation laws. Computer entropy condition linearization guided by both velocity Methods in Applied Mechanics and Engineering, 280, 57–83. and pressure. Compared with single entropy condition https://doi.org/10.1016/j.cma.2014.07.019 only guided by velocity, the new method is more accu- Harten, A. (1983). High resolution schemes for hyperbolic rate and reliable. It remedies the accuracy loss caused conservation laws. Journal of Computational Physics, 49(3), 357–393. https://doi.org/10.1016/0021-9991(83)90136-5 by rough prediction of previous method (especially for Henrick, A. K., Aslam, T. D., & Powers, J. M. (2005). Mapped multi-dimensional problems using Strang split). More- weighted essentially non-oscillatory schemes: Achieving over, compared with WENO + RK which needs entropy optimal order near critical points. Journal of Computational xfi for Roe flux (or using LLF flux which has more Physics, 207(2), 542–567. https://doi.org/10.1016/j.jcp.2005. numerical viscosity) to avoid non-physical solutions, 01.023 Full-WENO embedded with entropy condition automat- Huang, C-S, & Arbogast, T. (2017). An eulerian–lagrangian weighted essentially nonoscillatory scheme for nonlinear ically avoids non-physical solutions without extra artifi- conservation laws. Numerical Methods for Partial Dieff ren- cial numerical viscosity. tial Equations, 33(3), 651–680. https://doi.org/10.1002/num. The solution formula method is valuable for engi- neering applications, particularly for long-time evolution Huang, C-S, Arbogast, T., & Hung, C-H. (2016). A semi- problems. The entropy condition linearization is a criti- Lagrangian finite dieff rence WENO scheme for scalar nonlinear conservation laws. Journal of Computational cal technique in solution formula method which largely Physics, 322(4), 559–585. https://doi.org/10.1016/j.jcp.2016. determines the resolution and robustness in nonlinear 06.027 situations. However, there are still some flaws in current Jiang, G., & Shu, C. W. (1996). Efficient implementation of work: (1) How to remove the influence of numerical dis- weighted ENO schemes. Journal of Computational Physics, turbances on the entropy condition selection which may 126(1), 202–228. https://doi.org/10.1006/jcph.1996.0130 result in a resolution loss due to incorrectly choose the Lax, P. (1971). Shock waves and entropy. In E. H. Zarantonello (Ed.), Contributions to nonlinear functional analysis (pp. Roe means; (2) How to exactly predict when the com- 603–634). Academic Press. https://doi.org/10.1016/B978-0- pression waves evolve into shocks (we can also apply the 12-775850-3.50018-2 high-order flux reconstruction for compression waves Lax, P., & Wendro,ff B. ( 1960). Systems of conservation laws. before they evolve into shocks), etc. By solving these, Communications on Pure and Applied Mathematics, 13, the quality and robustness of numerical solutions may 217–237. https://doi.org/10.1002/cpa.3160130205 Li, J., & Du, Z. (2016). A Two-stage fourth order time-accurate be lifted to a new level. Therefore, in the future, we plan discretization for Lax-wendroff type flow solvers I. Hyper- to optimize the solution formula method from these two bolic conservation laws. SIAM Journal on Scientific Comput- aspects. Moreover, we also hope to develop this into a ing, 38(5), A3046–A3069. https://doi.org/10.1137/15M10- more efficient large time step scheme, which has been simply tried in (Zhou & Dong, 2022). If realized, it may Osher, S., & Harten, A. (1987). Uniformly high-order accu- greatly improve engineering ecffi iency. rate nonoscillatory schemes. I. SIAM Journal on Numerical Analysis, 24(2), 279–309. https://doi.org/10.1137/0724022 Note Qiu, J. (2007). WENO schemes with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations. Journal of Disclosure statement Computational and Applied Mathematics, 200(2), 591–605. https://doi.org/10.1016/j.cam.2006.01.022 No potential conflict of interest was reported by the author(s). Shen,Y., Liu, L.,&Yang,Y.(2014). Multistep weighted essentially non-oscillatory scheme. International Journal for References Numerical Methods in Fluids, 75(4), 231–249. https://doi.org/ 10.1002/fld.3889 Balsara, D. S., & Shu, C-W. (2000). Monotonicity preserving Shen,Y.,&Zha,G.(2014). Improvement of weighted essentially weighted essentially Non-oscillatory schemes with increas- non-oscillatory schemes near discontinuities. Computers ingly high order of accuracy. 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High order dis- cations, 50(1-2), 231–248. https://doi.org/10.1016/j.camwa. continuities decomposition entropy condition schemes for 2004.10.047 Euler equations. CFD Journal, 10(4), 448–457. 26 H. DONG ET AL. Strang, G. (1968). On the construction and comparison of dif- Methods in Applied Mechanics and Engineering, 382, 113853. ference schemes. SIAM Journal on Numerical Analysis, 5(3), https://doi.org/10.1016/j.cma.2021.113853 506–517. https://doi.org/10.1137/0705041 Zhao, J., Özgen-Xian, I., Liang, D., Wang, T., & Hinkelmann, R. Svenn Tveit. (2011). Numerical methods for conservation laws (2019). An improved multislope MUSCL scheme for solv- with a discontinuous u fl x function [Master]. ing shallow water equations on unstructured grids. Com- Sweby, P. K. (1984). High resolution schemes using u fl x limiters puters & Mathematics with Applications, 77(2), 576–596. for hyperbolic conservation laws. SIAM Journal on Numeri- https://doi.org/10.1016/j.camwa.2018.09.059 cal Analysis, 21(5), 995–1011. https://doi.org/10.1137/0721- Zhou, T., & Dong, H. T. (2021). A fourth-order entropy con- 062 dition scheme for systems of hyperbolic conservation laws. Titarev, V. A., & Toro, E. F. (2002). Ader: Arbitrary high Engineering Applications of Computational Fluid Mechanics, order Godunov approach. Journal of Scientific Comput- 15(1), 1259–1281. https://doi.org/10.1080/19942060.2021. ing, 17(1/4), 609–618. https://doi.org/10.1023/A:101512681 1955010 4947 Zhou, T., & Dong, H. T. (2022). A sixth order entropy condi- Toro,E.F.(2009). Riemann solvers and numerical methods for tion scheme for compressible flow. Computers & Fluids, 243, fluid dynamics (3rd ed.). Springer. 105514. https://doi.org/10.1016/j.compu fl id.2022.105514 Toro,E.F., &Hidalgo,A.(2009). ADER finitevolumeschemes Zorío, D., Baeza, A., & Mulet, P. (2017). An approxi- for nonlinear reaction–diffusion equations. Applied Numer- mate Lax–wendroff-type procedure for high order accurate ical Mathematics, 59(1), 73–100. https://doi.org/10.1016/j. schemes for hyperbolic conservation laws. Journal of Sci- apnum.2007.12.001 entific Computing , 71(1), 246–273. https://doi.org/10.1007/ Wang,Z., Zhu, J.,Yang, Y.,&Zhao,N.(2021). A new s10915-016-0298-2 fifth-order alternative finite difference multi-resolution WENO scheme for solving compressible flow. Computer ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 27 Appendices Appendix A: Discretization process of Newton Interpolation in Full-WENO We give the detailed Newton interpolation process for both initial value reconstruction and ux fl reconstruction of solution formula method.ConsiderFull-WENO5, forinitial valuereconstructionincase a ≥ 0, there are three third-order Newton interpolations of υ at [x , x ]withsix data around it,the middle oneisasfollows 1 1 j− j+ 2 2 υ −υ υ −υ 3 1 1 1 j+ j+ j+ j− 2 2 2 2 x −x x −x υ −υ 3 1 1 1 1 1 j+ j+ j+ j− j+ j− n 2 2 2 2 2 2 υ(x) = υ + (x − x ) + (x − x )(x − x ) 1 1 1 1 x −x 1 1 j+ j− j+ j+ x − x 2 3 1 2 2 j+ j− j+ j− 2 2 2 2 υ −υ υ −υ υ −υ υ −υ 3 1 1 1 1 1 1 3 j+ j+ j+ j− j+ j− j− j− 2 2 2 2 2 2 2 2 − − x −x x −x x −x x −x 3 1 1 1 1 1 1 3 j+ j+ j+ j− j+ j− j− j− 2 2 2 2 2 2 2 2 x −x x −x 3 1 1 3 j+ j− j+ j− 2 2 2 2 + × (x − x )(x − x )(x − x ) 3 1 1 x −x j− j− j+ 3 3 2 2 2 j+ j− 2 2 u −u u −u j+1 j j j−1 u −u n j+1 j 2h 2h = υ + u (x − x ) + (x − x )(x − x ) + (x − x )(x − x )(x − x ) 1 1 1 3 1 1 1 2h j+ j− j+ j− j− j+ j+ 3h 2 2 2 2 2 2 j− = υ + u (x − x ) + (x − x )(x − x ) + (x − x )(x − x )(x − x ) (A1) j 1 1 1 3 1 1 1 2 2h 3!h j+ j− j+ j− j− j+ j+ 2 2 2 2 2 2 We write the three Newton interpolations in square brackets and deduce the expression of u as follows j+ ⎡ ⎤ j− j−1 n 2 υ + u x − x + x − x x − x + x − x x − x x − x ⎢ 1 1 1 3 1 1 ⎥ j 2 1 2h 3!h j+ j− j+ j− j− j+ ⎢ j+ ⎥ 2 2 2 2 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ j− ⎢ n 2 ⎥ υ(x) = , υ + u x − x + x − x x − x + x − x x − x x − x 1 1 1 3 1 1 ⎢ j 2 ⎥ 1 2h 3!h j+ j− j+ j− j− j+ j+ ⎢ 2 2 2 2 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ j+ n j+1 υ + u x − x + x − x x − x + x − x x − x x − x j 1 1 1 1 1 3 1 2h 3!h j+ j− j+ j− j+ j+ j+ 2 2 2 2 2 2 1 1 x = (j + )h, x = (j + − ν)h ⇒ x − x = (2 − ν)h, x − x = (1 − ν)h, x − x = (−ν)h, 1 3 1 1 2 2 j+ j− j− j+ 2 2 2 2 x − x = (−1 − ν)h, j+ ⎡ ⎤ j− j−1 n 2 3 υ + u (−ν)h + (1 − ν)(−ν)h + (2 − ν)(1 − ν)(−ν)h ⎢ j ⎥ 1 2 2h 3!h ⎢ j+ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ j− n+1 1 j ⎢ n 2 3 ⎥ υ ≡ υ((j + − ν)h) = , υ + u (−ν)h + (1 − ν)(−ν)h + (2 − ν)(1 − ν)(−ν)h 1 j 2 2 ⎢ 1 ⎥ 2h 3!h j+ j+ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ j+ j+1 n 2 2 3 υ + u (−ν)h + (1 − ν)(−ν)h + (1 − ν)(−ν)(−1 − ν)h j 2 2h 3!h j+ ⎡ ⎤ ⎡ ⎤ 2−ν 1−ν 1 2 u + + j− 1 j−1 j−1 2 ⎢ 3 2 ⎥ j− u + (1 − ν) + (2 − ν)(1 − ν) ⎢ 2 3! ⎥ ⎢ ⎥ n+1 υ − υ ⎢ ⎥ ⎢ ⎥ 1 1 ⎢ ⎥ ⎢ ⎥ j+ j+ 2 2 2−ν 1−ν ⎢ j− ⎥ ⎢ ⎥ u ¯ ≡ = j = u + + .(A2) 1 2 1 ⎢ ⎥ ⎢ j 3 2 ⎥ u + (1 − ν) + (2 − ν)(1 − ν) j+ j− νh 2 2 3! 2 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ 1 2 ⎣ ⎦ j+ 2 −1−ν 1−ν j+1 u + + j 1 u + (1 − ν) + (1 − ν)(−1 − ν) j j+1 3 2 2 3! j+ 2 28 H. DONG ET AL. For a ≥ 0, there is one fifth-order Newton interpolation at [ x , x ] with six data around it 1 j 1 j− j+ 2 2 j− n 2 υ(x) = υ + u (x − x ) + (x − x )(x − x ) + (x − x )(x − x )(x − x ) 1 1 1 3 1 1 j 2 2h 3!h j+ j− j+ j− j− j+ j+ 2 2 2 2 2 2 j+ + (x − x )(x − x )(x − x )(x − x ) + (x − x )(x − x )(x − x )(x − x )(x − x ), 3 1 1 3 5 3 1 1 3 3 4 4!h 5!h j− j− j+ j+ j− j− j− j+ j+ 2 2 2 2 2 2 2 2 2 1 1 x = (j + )h, x = (j + − ν)h ⇒ x − x = (3 − ν)h, x − x = (2 − ν)h, x − x = (1 − ν)h, 1 5 3 1 2 2 j+ j− j− j− 2 2 2 2 x − x = (−ν)h, x − x = (−1 − ν)h, x − x = (−2 − ν)h, 1 3 5 j+ j+ j+ 2 2 2 j− n+1 1 n j 2 2 3 υ ≡ υ((j + − ν)h) = υ + u (−ν)h + (1 − ν)(−ν)h + (2 − ν)(1 − ν)(−ν)h 1 2 1 2h 3!h j+ j+ 2 2 j+ 4 5 + (2 − ν)(1 − ν)(−ν)(−1 − ν)h + (3 − ν)(2 − ν)(1 − ν)(−ν)(−1 − ν)h , 3 4 4!h 5!h n n+1 υ −υ 1 1 j+ j+ 3−ν −1−ν 2−ν 1−ν 2 2 2 3 4 u ¯ ≡ = u + ( + ( + ( + ) ) ) . 1 1 j 1 j νh 5 4 3 2 j+ j− j− 2 2 (A3) For ux fl reconstruction in case a ≥ 0, we need the derivatives of three third-order Newton interpolations as follows ⎡ ⎤ j− j−1 d d d u x − x + x − x x − x + x − x x − x x − x j 1 1 1 3 1 1 ⎢ 2 ⎥ dx 2h dx dx 3!h j+ j− j+ j− j− j+ ⎢ 2 2 2 2 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ j− ⎥ d d d u(x) = υ (x) = , x ⎢ ⎥ u x − x + x − x x − x + x − x x − x x − x j 1 1 1 3 1 1 dx 2h dx dx ⎢ 3!h ⎥ j+ j− j+ j− j− j+ 2 2 2 2 2 2 ⎢ ⎥ ⎢ ⎥ ⎣ 2 ⎦ j+ j+1 d d d u x − x + x − x x − x + x − x x − x x − x j 1 1 1 2 1 1 3 dx 2h dx dx j+ j− j+ 3!h j− j+ j+ 2 2 2 2 2 2 ⎡ ⎤ j− j−1 d 2 d d 2 ⎢ u (−(−ν)) + (−(1 − ν)(−ν))h + (−(2 − ν)(1 − ν)(−ν))h ⎥ j 2 dv 2h dv 3!h dv ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ n+1 j− u ≡ u(j + − ν) = j , ⎢ d 2 d d 2 ⎥ u (−(−ν)) + (−(1 − ν)(−ν))h + (−(2 − ν)(1 − ν)(−ν))h j 2 j+ ⎢ ⎥ dv 2h dv 3!h dv ⎢ ⎥ ⎣ ⎦ j+ j+1 d 2 d d 2 u (−(−ν)) + (−(1 − ν)(−ν))h + (−(1 − ν)(−ν)(−1 − ν))h j 2 dv 2h dv 3!h dv ⎡ ⎤ d −(−v)(1−v) 2 d −(−v)(1−v)(2−v) u + + j 1 dv 2! j−1 dv 3! j− ⎢ ⎥ ⎢ ⎥ d −(−v)(1−v) d −(−v)(1−v)(2−v) ⎢ 2 ⎥ u + + = j .(A4) ⎢ j ⎥ dv 2! dv 3! j− ⎢ ⎥ ⎣ ⎦ d −(−v)(1−v) 2 d −(−1−v)(−v)(1−v) u + + j 1 dv 2! j+1 dv 3! j+ And the expansion form of ux fl reconstruction in Equation (19) (for Full-WENO5) can be given as follows ⎡ ⎤ −2v+1 2 3v −6v+2 ⎪ u + + j 1 ⎪ j−1 2! 3! j− ⎢ 2 ⎥ ⎢ −2v+1 2 3v −6v+2 ⎥ u + + ⎪ j 1 , a ≥ 0, ⎢ ⎥ 2! j 3! j− ⎡ ⎤ ⎪ ⎣ 2 ⎦ ∗ ⎪ ⎪ 2 −2v+1 3v −1 ⎪ 2 ⎨ u + + j+1 2! 3! j+ ⎢ ⎥ u ≡ ⎡ ⎤ (A5) ⎣ ⎦ 2 −2v−1 2 3v −1 u + + ⎪ j+1 1 2! 3! ∗ ⎪ j+ u ⎢ 2 ⎥ ⎢ −2v−1 2 3v +6v+2⎥ ⎪ u + + j+1 3 , a ≤ 0. ⎢ ⎥ ⎪ 2! j+1 3! j+ ⎣ 2 ⎦ ⎪ −2v−1 3v +6v+2 u + + j+1 j+2 2! 3! j+ 2 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 29 The expansion form of flux reconstruction in Equation (20) (for Full-WENO7) can be given as follows ⎧ ⎡ ⎤ 2 3 2 −2v+1 2 3v −6v+2 3 −4v +18v −22v+6 u + + + j 1 2! j−1 3! 3 4! ⎪ j− ⎢ j− ⎥ ⎪ 2 ⎪ ⎢ ⎥ 2 3 2 ⎪ −2v+1 3v −6v+2 3 −4v +6v −2v−2 ⎪ ⎢ ⎥ u + + + j 1 2! j 3! 1 4! ⎢ ⎥ ⎪ j− j− ⎪ 2 ⎢ 2 ⎥ , a ≥ 0, 2 3 2 ⎪ ⎢ −2v+1 3v −6v+2 −4v +6v −2v−2 ⎥ 2 3 u + + + ⎢ 1 ⎥ ⎪ j 2! j 3! 1 4! ⎡ ⎤ ⎪ j− ⎢ j+ ⎥ ∗ ⎪ 2 ⎪ 2 ⎪ ⎣ ⎦ 1 2 3 2 ⎪ −2v+1 3v −1 −4v −6v +2v+2 2 3 u + + + ⎢ ⎥ ⎪ j j+1 3 ∗ 2! 3! 4! ⎨ j+ u j+ ⎢ ⎥ 2 2 ⎢ ⎥ ≡ (A6) ⎡ ⎤ ⎢ ⎥ ∗ 2 3 2 ⎪ −2v−1 3v −1 −4v +6v −2v−2 u 2 3 ⎣ ⎦ ⎪ 3 u + + + ⎪ j+1 j 1 ⎪ 2! 3! 4! j+ ⎪ j− ⎢ ⎥ ∗ ⎪ 2 u ⎪ ⎢ ⎥ 2 3 2 4 ⎪ −2v−1 2 3v +6v+2 3 −4v −6v +2v+2 ⎪ ⎢ ⎥ u + + + j+1 3 j+1 1 ⎪ 2! 3! 4! ⎢ j+ ⎥ ⎪ j+ ⎢ 2 ⎥ ⎪ , a ≤ 0. 2 3 2 ⎪ ⎢ ⎥ −2v−1 2 3v +6v+2 3 −4v −6v +2v+2 ⎪ u + + + ⎢ j+1 3 ⎥ j+1 3 ⎪ 2! 3! 4! j+ ⎪ j+ ⎢ ⎥ ⎣ 2 3 2 ⎦ −2v−1 2 3v +6v+2 3 −4v −18v −22v−6 ⎩ u + + + j+1 3 2! j+2 3! 5 4! j+ j+ Appendix B: Runge-Kutta method used in WENO-RK TVD third-order three-stage RK method n+1 n 1 1 4 u = L(u), u = u + τ( L + L + L ) t 1 2 3 6 6 6 L ≡ L(u ), n (B1) ⎪ L ≡ L(u + τL ), n + 1 2 1 1 1 1 L ≡ L(u + τ( L + L )), n + 3 1 2 4 4 2 Nystrom fifth-order six-stage RK method n+1 n 23 125 −81 125 u = L(u), u = u + τ( L + L + L + L ) t 1 3 5 6 ⎪ 192 192 192 192 L ≡ L(u ), n n τ 1 ⎪ L ≡ L(u + L ), n + 2 1 ⎪ 3 3 4 6 2 L ≡ L(u + τ( L + L )), n + (B2) 3 1 2 25 25 5 n 1 −12 15 L ≡ L(u + τ( L + L + L )), n + 1 4 1 2 3 4 4 4 n 6 90 −50 8 2 ⎪ L ≡ L(u + τ( L + L + L + L )), n + 5 1 2 3 4 81 81 81 81 3 6 36 10 8 4 L ≡ L(u + τ( L + L + L + L )), n + 6 1 2 3 4 75 75 75 75 5 Butcher seventh-order nine-stage RK method n+1 n 32 1771561 243 16807 77 11 u = L(u), u = u + τ( L + L + L + L + L + L ) ⎪ t 4 5 6 7 8 9 105 6289920 2560 74880 1440 270 ⎪ L ≡ L(u ), n n τ 1 ⎪ L ≡ L(u + L ), n + 2 1 ⎪ 6 6 n τ 1 L ≡ L(u + L ), n + 3 2 ⎪ 3 3 1 3 1 L ≡ L(u + τ( L + L )), n + 4 1 3 8 8 2 (B3) 148 150 −56 2 L ≡ L(u + τ( L + L + L )), n + 5 1 3 4 ⎪ 1331 1331 1331 11 ⎪ −404 −170 4024 10648 2 L ≡ L(u + τ( L + L + L + L )), n + ⎪ 6 1 3 4 5 243 27 1701 1701 3 ⎪ n 2466 1242 −19176 −51909 1053 6 L ≡ L(u + τ( L + L + L + L + L )), n + ⎪ 7 1 3 4 5 6 2401 343 16807 16807 2401 7 ⎪ n 5 96 −1815 −405 49 L ≡ L(u + τ( L + L + L + L + L )), n ⎪ 8 1 4 5 6 7 154 539 20384 2464 1144 n −113 −195 32 29403 −729 1029 21 L ≡ L(u + τ( L + L + L + L + L + L + L )), n + 1 9 1 3 4 5 6 7 8 32 22 7 3584 512 1408 16 According to (Butcher, 2016): above order 4, it is no longer possible to obtain order s with just s stages. For order 5, six stages are required, and for order 7, nine stages are required.

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Engineering Applications of Computational Fluid Mechanics – Taylor & Francis

Published: Jan 1, 2

Keywords: solution formula method; fully discrete; full-WENO; Euler equations; hyperbolic conservation laws; unsteady compressible flow

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Fully discrete WENO with double entropy condition for hyperbolic conservation laws

Fully discrete WENO with double entropy condition for hyperbolic conservation laws

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Fully discrete WENO with double entropy condition for hyperbolic conservation laws

Fully discrete WENO with double entropy condition for hyperbolic conservation laws

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