High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

In this paper,we propose a high-order accurate discontinuous Galerkin(DG)method for the compressible Euler equations under gravitationalfields on un-structured meshes.The scheme preserves a general hydrostatic equilibrium state and provably guarantees the positivity of density and pressure at the same time.Compar-ing with the work on the well-balanced scheme for Euler equations with gravitation on rectangular meshes,the extension to triangular meshes is conceptually plausible but highly nontrivial.Wefirst introduce a special way to recover the equilibrium state and then design a group of novel variables at the interface of two adjacent cells,which plays an important role in the well-balanced and positivity-preserving properties.One main challenge is that the well-balanced schemes may not have the weak positivity property.In order to achieve the well-balanced and positivity-preserving properties simultaneously while maintaining high-order accuracy,we carefully design DG spa-tial discretization with well-balanced numericalfluxes and suitable source term ap-proximation.For the ideal gas,we prove that the resulting well-balanced scheme,cou-pled with strong stability preserving time discretizations,satisfies a weak positivity property.A simple existing limiter can be applied to enforce the positivity-preserving property,without losing high-order accuracy and conservation.Extensive one-and two-dimensional numerical examples demonstrate the desired properties of the pro-posed scheme,as well as its high resolution and robustness.