We have developed a Runge-Kutta discontinuous Galerkin (RKDG) method for solving wave equations in isotropic and anisotropic poroelastic media at low frequencies. First, the 2D Biot’s two-phase equations are transformed into a first-order system with dissipation. Then, the system is discretized by using the discontinuous Galerkin method with a third-order Runge-Kutta time discretization. The numerical stability conditions for solving porous equations are also investigated. We test several examples to validate our method in isotropic and anisotropic poroelastic media. Comparisons of seismic responses with the finite-difference method on fine grids show the correctness of this method. Moreover, the numerical results indicate that the RKDG method can provide clear fast P-, slow P-, and S-waves for anisotropic poroelastic media on coarse meshes. Also, a two-layer porous model, a poroelastic-elastic model with horizontal interface, and an isotropic-anisotropic poroelastic model with sinusoidal interface demonstrate that our method can deal with complex wave propagation. Therefore, the simulation results show that RKDG is an accurate and stable method for solving Biot’s equations.

中文翻译：

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用于求解二维各向同性和各向异性多孔弹性介质中低频波动方程的 Runge-Kutta 不连续伽辽金方法

我们开发了一种 Runge-Kutta 不连续伽辽金 (RKDG) 方法，用于求解低频下各向同性和各向异性多孔弹性介质中的波动方程。首先，将二维 Biot 的两相方程转化为具有耗散的一阶系统。然后，使用具有三阶 Runge-Kutta 时间离散化的不连续 Galerkin 方法对系统进行离散化。还研究了求解多孔方程的数值稳定性条件。我们测试了几个例子来验证我们在各向同性和各向异性多孔弹性介质中的方法。在细网格上与有限差分法的地震响应比较表明了该方法的正确性。此外，数值结果表明，RKDG 方法可以为粗网格上的各向异性多孔弹性介质提供清晰的快速 P 波、慢速 P 波和 S 波。还，两层多孔模型、具有水平界面的多孔弹性-弹性模型和具有正弦界面的各向同性-各向异性多孔弹性模型表明我们的方法可以处理复杂的波传播。因此，仿真结果表明 RKDG 是求解 Biot 方程的一种准确且稳定的方法。