Two robust weighted Runge-Kutta discontinuous Galerkin methods for scalar wavefield modeling

Abstract

Developing fast, efficient forward modeling methods with low numerical dispersion is an important research topic in computational geophysics. The discontinuous Galerkin (DG) method shows advantages in suppressing numerical dispersion and supporting flexible meshing, but it faces the disadvantage of slow computation. To accelerate the simulation, one strategy is to relax the CFL condition, thereby allowing larger time steps. Based on the weighted Runge-Kutta discontinuous Galerkin (WRK-DG) method, in which WRK is the time discretization scheme and DG the spatial discretization scheme, we propose two robust WRK time discretization schemes: the coefficient-optimized WRK scheme and the stepwise computation-reduced WRK scheme. These schemes are incorporated into the original DG spatial framework for the efficient solution of the scalar wave equation. In addition, our proposed methods include both the numerical flux-based DG method and the penalty function-based DG method. We establish the numerical stability conditions of these schemes and analyze the numerical dispersion characteristics for P¹–P⁵ elements on rectangular meshes. The results show that the two new methods increase the maximum CFL condition numbers and can effectively suppress numerical dispersion. For the numerical flux-based DG method, compared with the third-order RKDG method, the CFL condition number of the coefficient-optimized WRK-DG method is more than three times larger, and the CFL number of the stepwise computation-reduced WRK-DG method is 1.68–2.1 times larger. For the penalty-based DG method, the coefficient-optimized WRK-DG scheme yields a CFL condition number approximately 2.5 times larger than the third-order RKDG method, while the stepwise computation-reduced WRK-DG scheme achieves CFL numbers about 2.16 times larger. Meanwhile, our numerical experiments demonstrate that the proposed DG methods can greatly improve the computational efficiency.