High order structure-preserving arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the Euler equations under gravitational fields

In this work, we present high-order arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods for the Euler equations under gravitational fields on the moving mesh. The goal of this paper is to demonstrate that, through careful design of the scheme, the ALE-DG methods can also achieve the structure-preserving properties of DG methods, such as high order accuracy, well-balanced property, positivity-preserving property, for the Euler equations with arbitrary moving meshes. We propose two well-balanced and positivity-preserving ALE-DG schemes which can preserve the explicitly given equilibrium state on arbitrary moving grids, and also carry out rigorous positivity-preserving analyses for both schemes. Our schemes are established both in one dimension and in two dimensions on unstructured triangular meshes. The most challenging component in designing such ALE-DG schemes on the moving mesh is to maintain the equilibrium state and the mass conservation at the same time, since temporal discretization of the ALE method may destroy the well-balanced property, and inappropriate adjustment of the numerical flux could lead to the loss of the mass conservation property on the moving meshes. A novel approximation of the desired equilibrium state based on ALE-DG methods on the moving mesh has been introduced to overcome such difficulty. Numerical experiments in different circumstances are provided to illustrate the well-balanced property, positivity-preserving property and high order accuracy. We also compare the schemes on the moving mesh and on the static mesh to demonstrate the advantage of ALE-DG methods for discontinuous solutions.