High order well-balanced Arbitrary-Lagrangian-Eulerian ADER discontinuous Galerkin schemes on general polygonal moving meshes

Abstract

In this work, we present a novel family of high order accurate numerical schemes for the solution of hyperbolic partial differential equations (PDEs) which combines several geometrical and physical structure preserving properties. First, we settle our methods in the Lagrangian framework, where each element of the mesh evolves following as close as possible the local fluid flow, so to reduce the numerical dissipation at contact waves and moving interfaces and to satisfy the Galilean and rotational invariance properties of the studied PDEs system. In particular, we choose the direct Arbitrary-Lagrangian-Eulerian (ALE) approach which, in order to always guarantee the high quality of the moving mesh, allows to combine the Lagrangian motion with mesh optimization techniques. The employed polygonal tessellation is thus regenerated at each time step, the previous one is connected with the new one by spacetime control volumes, including hole-like sliver elements in correspondence of topology changes, over which we integrate a spacetime divergence form of the original PDEs through a high order accurate ADER discontinuous Galerkin (DG) scheme. Mass conservation and adherence to the GCL condition are guaranteed by construction thanks to the integration over closed control volumes, and robustness over shock discontinuities is ensured by the use of an a posteriori subcell finite volume (FV) limiting technique. On top of this effective moving mesh framework, we have also modified the full ADER-DG scheme with a posteriori subcell FV limiter to be, for the first time in literature, well-balanced. This is achieved by ensuring that any projection, reconstruction and integration procedures were always performed by summing up the exact value of a given equilibrium plus the high order accurate evolution of the fluctuations w.r.t. said equilibrium. The paper is closed by a wide set of numerical results, including simulations of Keplerian disks, which demonstrate all the claimed properties and the increased accuracy of our novel family of schemes, in particular for the evolution of small perturbations arising over moving equilibrium profiles.