Runge–Kutta discontinuous Galerkin method based on flux vector splitting for hyperbolic conservation laws

Abstract

The flux vector splitting (FVS) method has firstly been incorporated into the Runge–Kutta Discontinuous Galerkin (RKDG) framework for reconstructing the numerical fluxes required for the spatial semi-discrete formulation, setting it apart from the conventional RKDG approaches that typically utilize the Lax–Friedrichs flux scheme or classical Riemann solvers such as HLLC. The control equations are initially reformulated into a flux-split form. Subsequently, a variational approach is applied to this flux-split form, from which a DG spatial semi-discrete scheme based on FVS is derived. Then, FVS-RKDG is implemented in two-dimensional case by splitting the normal flux on cell interfaces instead of splitting dimension by dimension in the x and y directions Finally, the concept of “flux vector splitting based on Jacobian eigenvalue decomposition” has been applied to the conservative linear scalar transport equations and the nonlinear Burgers’ equation. This approach has led to the rederivation of the classical Lax–Friedrichs flux scheme and the provision of a Steger–Warming flux scheme for scalar cases.