A Discontinuous Galerkin method for three-dimensional elastic and poroelastic wave propagation: forward and adjoint problems

We develop a numerical solver for three-dimensional wave propagation in coupled poroelastic-elastic media, 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, including the poroelastic-elastic interface; 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.