An effective discontinuous Galerkin method for solving acoustic wave equations in heterogeneous media

Abstract A discontinuous Galerkin (DG) method is introduced for solving 2D and 3D first-order velocity-pressure acoustic wave propagation problems in heterogeneous media. First, acoustic equations are reformulated into a first-order hyperbolic system, which can be integrated into the framework of the DG method. Then, we introduce an effective numerical flux in DG spatial discretization, which preserves physical laws on both sides of interfaces with discontinuous material parameters. It is compared with both the classic Lax–Friedrichs flux and the exact upwind flux, with the latter derived from solving the Riemann problem and satisfying the Rankine–Hugoniot condition. The comparison reveals that while our flux is formally similar to the classic local Lax–Friedrichs flux, its numerical behavior is analogous to the exact upwind flux. The primary advantage of this method lies in its simplicity and straightforward computation yet can maintain physical continuity conditions near interfaces with discontinuous material parameters. Several numerical experiments of acoustic wave propagation in 2D and 3D heterogeneous media are carried out, illustrating the effectiveness of this method.