Uniformly High-Order Structure-Preserving Discontinuous Galerkin Methods for Euler Equations with Gravitation: Positivity and Well-Balancedness

This paper presents a class of novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature these is that they are well-balanced general hydrostatic equilibrium state, and at same time, provably preserve positivity density pressure. In order to achieve positivity-preserving properties simultaneously, DG spatial discretization carefully designed with suitable source term reformulation properly modified Harten-Lax-van Leer contact (HLLC) flux. Based on some technical decompositions as well several key admissible states HLLC flux, rigorous analyses carried out. It proven resulting schemes, coupled strong stability preserving time discretizations, satisfy weak property, which implies one can apply simple existing limiter effectively enforce without losing accuracy conservation. The proposed methods applicable system equation state. Extensive one- two-dimensional numerical tests demonstrate desired including exact preservation ability capture small perturbation such robustness solving problems involving low and/or pressure, good resolution smooth solutions.