Dispersion–dissipation analysis of the triangle-based discontinuous Galerkin method for scalar wave equation

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In this paper, we present a dispersion–dissipation analysis of the triangle-based discontinuous Galerkin method (DGM) for scalar wave equation. The DGM is based on the numerical flux formulations with high-order accuracy. The analysis includes a semi-discrete case and a fully discrete case. The semi-discrete analysis relies on the discrete eigenvalue problem for spatially propagating waves. Three types of regular triangular meshes are considered. In addition, we compare three commonly used numerical fluxes—the centred flux, the Riemann flux, and the local Lax-Friedrichs (LLF) flux. The fully discrete dispersion–dissipation analysis is integrated with two kinds of time discretization—an arbitrary high-order derivative (ADER) time discretization and a weighted Runge–Kutta time discretization. The results indicate that the numerical dispersion and dissipation behave differently for different mesh types, demonstrating the importance of considering the mesh types when generating grids for practical simulations. Also, it is demonstrated that under the same grid conditions, different numerical fluxes result in differences in the numerical dispersion and dissipation. This suggests the need for further research to determine the optimal numerical flux that achieves a balance between the numerical dispersion and dissipation or improves both. The fully discrete dispersion–dissipation analysis demonstrates that different time discretization schemes introduce different sizes of numerical dispersion and dissipation. Finally, numerical experiments are presented to verify some of the theoretical findings.