Uniformly high-order structure-preserving discontinuous galerkin methods for euler equations with gravitation: Positivity and well-balancedness

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