求解双相和黏弹性介质波传播方程的间断有限元方法及其波场模拟

间断有限元（Discontinuous Galerkin：DG）方法具有低数值频散、网格剖分灵活、能模拟地震波在复杂介质中传播等优点.因此，本文将一种新的DG方法推广到双相和黏弹性等复杂介质的地震波场模拟，发展了求解Biot弹性波方程和D'Alembert介质波动方程的DG方法.首先通过引入辅助变量将Biot双相介质弹性波方程和D'Alembert介质波动方程转化为关于时间-空间的一阶偏微分方程组，然后对该方程组进行DG空间离散，得到半离散化的常微分方程组.最后，对此常微分方程组，应用加权的Runge-Kutta格式进行时间推进计算.数值结果表明，DG方法可以有效地求解Biot双相介质弹性波方程和D'Alembert介质波动方程，并能很好地压制因离散求解波动方程而产生的数值频散，获得清晰的各种地震波震相.

Discontinuous Galerkin method for solving wave equations in two-phase and viscoelastic media

The Discontinuous Galerkin (DG) method has great advantages in suppressing numerical dispersion and dealing with complex structures. Therefore, in this paper, we apply a new DG method to numerical simulations in two-phase and viscoelastic media, and suggest a DG method to solve both Biot elastic wave equations and the D'Alembert wave equations. For this, we first transform the Biot equations and the D'Alembert wave equations into a system of first-order equations with respect to time-space by introducing auxiliary variables. Then we transform the first-order equations into a semi-discrete ordinary differential equation (ODE) system using the DG method. Finally, we use a weighted Runge-Kutta method to solve the ODE system. The numerical results show that the DG method works very well for solving the Biot elastic wave equations and D'Alembert wave equations, and can effectively suppress the numerical dispersion and provide accurate information on the wave-field.