The positivity preserving property on the high order arbitrary Lagrangian-Eulerian discontinuous Galerkin method for Euler equations

Abstract

This paper presents an almost arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve the compressible Euler equations in one and two space dimensions, and consider their positivity preserving property of states density and pressure. The ALE-DG method coupling with a modified strong stability-preserving Runge-Kutta method is developed to ensure the geometric conservation law and the positivity property of the scheme with the positivity preserving limiter developed by Zhang et al. (2010) [34] and Zhang et al. (2012) [38]. For the Lax-Friedrichs, HLL, and HLLC numerical fluxes, we prove that our proposed ALE-DG method can keep the positivity property of approximations of density and pressure from the first-order ALE-DG method and the cell averages of approximations of density and pressure from high order ALE-DG method under suitable time step. Numerical examples demonstrate that the proposed positivity-preserving ALE-DG method can keep the high order accuracy, numerical stability, and positivity on moving meshes for Euler equations.