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A Discontinuous Galerkin Method for Three-Dimensional Poroelastic Wave Propagation: Forward and Adjoint Problems

A Discontinuous Galerkin Method for Three-Dimensional Poroelastic Wave Propagation: Forward and... We develop a numerical solver for three-dimensional poroelastic wave propagation, based on a high-order discontinuous Galerkin (DG) method, with the Biot poroelastic wave equation formulated as a first order conservative velocity/strain hyperbolic system. To derive an upwind numerical flux, we find an exact solution to the Riemann problem; we also consider attenuation mechanisms both in Biot’s low- and high-frequency regimes. Using either a low-storage explicit or implicit–explicit (IMEX) Runge–Kutta scheme, according to the stiffness of the problem, we study the convergence properties of the proposed DG scheme and verify its numerical accuracy. In the Biot low frequency case, the wave can be highly dissipative for small permeabilities; here, numerical errors associated with the dissipation terms appear to dominate those arising from discretisation of the main hyperbolic system. We then implement the adjoint method for this formulation of Biot’s equation. In contrast with the usual second order formulation of the Biot equation, we are not dealing with a self-adjoint system but, with an appropriate inner product, the adjoint may be identified with a non-conservative velocity/stress formulation of the Biot equation. We derive dual fluxes for the adjoint and present a simple but illuminating example of the application of the adjoint method. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

A Discontinuous Galerkin Method for Three-Dimensional Poroelastic Wave Propagation: Forward and Adjoint Problems

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-021-00395-4
Publisher site
See Article on Publisher Site

Abstract

We develop a numerical solver for three-dimensional poroelastic wave propagation, based on a high-order discontinuous Galerkin (DG) method, with the Biot poroelastic wave equation formulated as a first order conservative velocity/strain hyperbolic system. To derive an upwind numerical flux, we find an exact solution to the Riemann problem; we also consider attenuation mechanisms both in Biot’s low- and high-frequency regimes. Using either a low-storage explicit or implicit–explicit (IMEX) Runge–Kutta scheme, according to the stiffness of the problem, we study the convergence properties of the proposed DG scheme and verify its numerical accuracy. In the Biot low frequency case, the wave can be highly dissipative for small permeabilities; here, numerical errors associated with the dissipation terms appear to dominate those arising from discretisation of the main hyperbolic system. We then implement the adjoint method for this formulation of Biot’s equation. In contrast with the usual second order formulation of the Biot equation, we are not dealing with a self-adjoint system but, with an appropriate inner product, the adjoint may be identified with a non-conservative velocity/stress formulation of the Biot equation. We derive dual fluxes for the adjoint and present a simple but illuminating example of the application of the adjoint method.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Dec 1, 2021

Keywords: Discontinuous Galerkin method; Poroelastic waves; Adjoint method; 86-08; 35R30

References