Access the full text.
Sign up today, get DeepDyve free for 14 days.
Claire David, Rasika Fernando, Zhaosheng Feng (2005)
On solitary wave solutions of the compound Burgers–Korteweg–de Vries equationPhysica A-statistical Mechanics and Its Applications, 375
Q. Lin, Aihui Zhou (1993)
CONVERGENCE OF THE DISCONTINUOUS GALERKIN METHOD FOR A SCALAR HYPERBOLIC EQUATIONActa Mathematica Scientia, 13
A. Polyanin (2004)
Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations , Chapman & Hall/CRC, Boca
J. Burgers (1995)
Mathematical Examples Illustrating Relations Occurring in the Theory of Turbulent Fluid Motion
M. Feistauer, Karel Svadlenka (2004)
Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems, 12
E. Ince
Ordinary differential equations
G. Whitham, T. Ting (1976)
Linear and Nonlinear Waves
Claire David, P. Sagaut (2008)
Structural stability of finite dispersion-relation preserving schemesChaos Solitons & Fractals, 41
Aihui Zhou (1991)
EXTRAPOLATION FOR COLLOCATION METHOD OF THE FIRST KIND VOLTERRA INTEGRAL EQUATIONSActa Mathematica Scientia, 11
P. Lesaint, P. Raviart (1974)
On a Finite Element Method for Solving the Neutron Transport Equation
Claire David, P. Sagaut (2006)
Spurious solitons and structural stability of finite-difference schemes for non-linear wave equationsChaos Solitons & Fractals, 41
R.S. Johnson (1997)
A Modern Introduction to the Mathematical Theory of Water Waves
E. Toro (1997)
Riemann Solvers and Numerical Methods for Fluid Dynamics
J. Burkill, G. Birkhoff, G. Rota (1964)
Ordinary Differential EquationsThe Mathematical Gazette, 48
Claes Johnson, J. Pitkäranta (1986)
An analysis of the discontinuous Galerkin method for a scalar hyperbolic equationMathematics of Computation, 46
Q. Lin, A. Zhou (1991)
Some arguments for recovering the finite element error of hyperbolic problemsActa Math. Sci., 11
A. Harten, P. Lax, B. Leer (1983)
On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation LawsSiam Review, 25
C. David, R. Fernando, Z. Feng (2007)
A note on “general solitary wave solutions of the compound Burgers-Korteweg-de Vries EquationPhys. A, Stat. Theor. Phys., 375
F. Giraldo, J. Hesthaven, T. Warburton (2002)
Nodal high-order discontinuous Galerkin methods for the spherical shallow water equationsJournal of Computational Physics, 181
E.F. Toro (1997)
Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction
P. Hartman (1965)
Ordinary Differential EquationsJournal of the American Statistical Association, 60
P. Lesaint, P. Raviart (1974)
Mathematical Aspects of Finite Elements in Partial Differential Equations
Q. Lin, Q. Zhu (1994)
The Preprocessing and Postprocessing for the Finite Element Methods
R. Johnson (1997)
A Modern Introduction to the Mathematical Theory of Water Waves: Bibliography
P. Houston, Max Jensen, E. Süli (2002)
hp-Discontinuous Galerkin Finite Element Methods with Least-Squares StabilizationJournal of Scientific Computing, 17
H. Atkins, Chin-Shu Wang (1996)
QUADRATURE-FREE IMPLEMENTATION OF DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONSAIAA Journal, 36
Bernardo Cockburn (2003)
Discontinuous Galerkin methodsZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 83
A. Polyanin, V. Zaitsev (2003)
Handbook of Nonlinear Partial Differential Equations
Bernardo Cockburn, Chi-Wang Shu (1989)
TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General frameworkMathematics of Computation, 52
Bernardo Cockburn, Chi-Wang Shu (2001)
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated ProblemsJournal of Scientific Computing, 16
R. Dodd, J. Eilbeck, J. Gibbon, H. Morris (1982)
Solitons and Nonlinear Wave Equations
The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a discontinuous Galerkin finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to discontinuous Galerkin schemes (David and Sagaut in Chaos Solitons Fractals 41(4):2193–2199, 2009; Chaos Solitons Fractals 41(2):655–660, 2009).
Acta Applicandae Mathematicae – Springer Journals
Published: Aug 18, 2010
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.