Discontinuous Galerkin Method For Solving Wave Equations

In order to satisfy the numerical simulation for complex geological structures, this dissertation introduces the discontinuous Galerkin(DG) method into seismic wave-field modeling. First, the classical DG method- Runge Kutta DG(RKDG) method is studied in detail on its numerical stability and numerical dispersion. It is also compared with other numerical methods in numerical dispersion and computational efficiency. Theoretical analyses and numerical tests show that RKDG method can suppress the numerical dispersion more effectively. However, the Coutant-Friedrichs-Lewy(CFL) numerical stability condition for RKDG method is stricter than other numerical methods, with lower computational efficiency and higher storage requirements, which is not suitable for large scale forward wave-field simulation, seismic migration and inversion imaging.In order to increase the computational efficiency of RKDG method, the first idea aims at improving its strict CFL condition numbers. The dissertation proposes a weighted Runge-Kutta discontinuous Galerkin(WRKDG) method. This method employs DG numerical flux formulations for spatial discretization, and employs an implicit diagonal Runge-Kutta method for its time discretization. A two- iterations procedure is introduced to convert the implicit scheme to an explicit one. In addition, a weighted factor is set in the iterative process to enrich the method. Then, WRKDG method is investigated in detail, including its numerical error, numerical stability condition and numerical dispersion. The analyses indicate that the CFL condition for WRKDG method is more relaxed compared with the classic RKDG method, without obvious increase in numerical dispersion. In particular, the maximal Courant number for P1 and P2 WRKDG methods are 1.096 and 0.338, respectively, resulting in a 3.5 times superiority for P1 element and a 2 times superiority for P2 element compared with TVD RKDG method. The analyses also indicate that WRKDG method can suppress numerical dispersion more efficiently than staggered-grid(SG) method. Finally, WRKDG method is applied to simulate several complex geologic models. Numerical results show that WRKDG method can provide accurate information on the wave field and effectively suppress numerical dispersion.The second idea to improve the computational efficiency of RKDG method is based on domain decomposition. The idea is to combine the DG method with other numerical method. A hybrid scheme based on finite difference method—optimal nearly analytic discrete(ONAD) method, and DG method—WRKDG method is developed. The hybrid scheme combines the advantages of both finite difference method and DG method, and avoids their disadvantages. Numerical experiments show that, the hybrid algorithm can suppress numerical dispersion efficiently and can significantly improve the computational efficiency. It’s especially suitable for the simulation of wave propagation in complex and multi-scale structure media.