A numerical dispersion-dissipation analysis of discontinuous Galerkin methods based on quadrilateral and triangular elements

Abstract

The dispersive and dissipative properties of numerical methods are important for numerical modeling. We have evaluated a numerical dispersion-dissipation analysis for two discontinuous Galerkin methods (DGMs) — the flux-based DGM (FDGM) and the interior penalty DGM (IP DGM) for scalar wave equation. The semidiscrete analysis based on the plane-wave analysis is conducted for quadrilateral and triangular elements. Two kinds of triangular elements are taken into account. The fully discrete analysis for each method is conducted by incorporating a classic third-order total variation diminishing (TVD) Runge-Kutta (RK) time discretization. Our results indicate that FDGM produces smaller numerical dispersion than IP DGM, but it introduces more numerical dissipation. Notably, the two methods have different local convergence orders for numerical dispersion and dissipation. The anisotropy properties for different mesh types can also be identified. Several numerical experiments are carried out that verify some theoretical findings. The experiments exhibit that the numerical error introduced by FDGM is less than that introduced by IP DGM whereas the storage and calculation time of FDGM are greater than that of IP DGM. Overall, our work indicates that when both methods adopt third TVD RK time discretization, the computational efficiency of FDGM is slightly larger than IP DGM.