A two-stage two-derivative fourth order positivity-preserving discontinuous Galerkin method for hyperbolic conservation laws

Abstract

In this paper, a fourth order positivity-preserving (PP) scheme for hyperbolic conservation laws based on the two-stage two-derivative fourth order ($S_2{D}_2{O}_4$) time discretization and discontinuous Galerkin (DG) spatial discretization is developed. We construct a local Lax–Friedrichs type PP flux in the sense that the DG scheme with this flux satisfies the PP property. We use the strong stability preserving (SSP) $S_2{D}_2{O}_4$ time discretization and obtain the PP conditions for one-dimensional scalar conservation laws. With a PP limiter introduced in Zhang and Shu (2010) [51], the SSP $S_2{D}_2{O}_4$ DG schemes are rendered preserving the positivity without losing conservation or high order accuracy. We carry out the extension of the method to two dimensions on rectangular meshes. Based on this idea, we further develop high-order DG schemes which can preserve the positivity of density and pressure for compressible Euler equations. Numerical tests for the fourth order DG schemes are reported to demonstrate the effectiveness of the algorithms.