High-Order Godunov-Type Scheme for Conservation Laws with Discontinuous Flux Function in Space

Abstract

The numerical method for the conservation laws with discontinuous flux in space is considered. The difficulty in designing a high-order accurate and maintaining an efficient well-balanced scheme for this system lies in the fact that the flux is discontinuous across the stationary discontinuity, which results in the jump of the unknown function. In order to overcome this difficulty, the Godunov-type numerical flux is written in the positive and negative fluxes for the first time, which is embedded between two discontinuous flux functions. With the Godunov flux-splitting, the high-order accurate scheme is proposed through the WENO reconstruction of flux and the third-order TVD Runge–Kutta time discretization. Some tests are presented to demonstrate that our new high-order scheme is capable of exactly preserving steady-state solutions.