Positivity-Preserving Well-Balanced Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Methods for the Shallow Water Equations

In this paper, we develop well-balanced arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods for the shallow water equations, which preserve not only the still water equilibrium but also the moving water equilibrium. Based on the time-dependent linear affine mapping, the ALE-DG method for conservation laws maintains almost all mathematical properties of DG methods on static grids, such as conservation, geometric conservation law (GCL), entropy stability. The main difficulty to obtain the well-balanced property of the ALE-DG method for shallow water equations is that the grid movement and usual time discretization may destroy the equilibrium. By adopting the GCL preserving Runge-Kutta methods and the techniques of well-balanced DG schemes on static grids, we successfully construct the high order well-balanced ALE-DG schemes for the shallow water equations with still and moving water equilibria. Meanwhile, the ALE-DG schemes can also preserve the positivity property at the same time. Numerical experiments in different circumstances are provided to illustrate the well-balanced property, positivity preservation and high order accuracy of these schemes.