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.