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.

我们基于高阶不连续伽辽金 (DG) 方法开发了一个用于三维多孔弹性波传播的数值求解器，Biot 多孔弹性波方程被公式化为一阶保守速度/应变双曲线系统。为了导出逆风数值通量，我们找到了黎曼问题的精确解；我们还考虑了 Biot 低频和高频区域的衰减机制。使用低存储显式或隐式-显式 (IMEX) Runge-Kutta 方案，根据问题的刚度，我们研究了所提出的 DG 方案的收敛特性并验证其数值精度。在 Biot 低频情况下，对于小磁导率，波可以是高耗散的；这里，与耗散项相关的数值误差似乎在主双曲系统离散化产生的误差中占主导地位。然后我们为 Biot 方程的这个公式实现伴随方法。与通常的 Biot 方程的二阶公式相比，我们不是在处理自伴随系统，但是，通过适当的内积，可以用 Biot 方程的非保守速度/应力公式识别伴随系统。我们推导出伴随法的双通量，并展示了一个简单但有启发性的应用伴随法的例子。使用适当的内积，可以使用 Biot 方程的非保守速度/应力公式来识别伴随。我们推导出伴随法的双通量，并展示了一个简单但有启发性的应用伴随法的例子。使用适当的内积，可以使用 Biot 方程的非保守速度/应力公式来识别伴随。我们推导出伴随法的双通量，并展示了一个简单但有启发性的应用伴随法的例子。