A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow

Abstract

The arbitrary Lagrangian-Eulerian (ALE) method is widely used in the field of compressible multi-material and multi-phase flow problems. In order to implement the indirect ALE approach for the simulation of compressible flow in the context of high order discontinuous Galerkin (DG) discretizations, we present a high order positivity-preserving DG remapping method based on a moving mesh solver in this paper. This remapping method is based on the ALE-DG method developed by Klingenberg et al. [17, 18] to solve the trivial equation $\frac{∂u}{∂t} = 0$ on a moving mesh, which is the old mesh before remapping at $t = 0$ and is the new mesh after remapping at $t = T.$ An appropriate selection of the final pseudo-time $T$ can always satisfy the relatively mild smoothness requirement (Lipschitz continuity) on the mesh movement velocity, which guarantees the high order accuracy of the remapping procedure. We use a multi-resolution weighted essentially non-oscillatory (WENO) limiter which can keep the essentially non-oscillatory property near strong discontinuities while maintaining high order accuracy in smooth regions. We further employ an effective linear scaling limiter to preserve the positivity of the relevant physical variables without sacrificing conservation and the original high order accuracy. Numerical experiments are provided to illustrate the high order accuracy, essentially non-oscillatory performance and positivity-preserving of our remapping algorithm. In addition, the performance of the ALE simulation based on the DG framework with our remapping algorithm is examined in one- and two-dimensional Euler equations.