A high order positivity-preserving conservative WENO remapping method based on a moving mesh solver

Abstract

In this paper, a high order accurate positivity-preserving conservative remapping algorithm is developed. Quadrilateral meshes in two dimensions are used as examples. This remapping method is based on the numerical solution of the trivial equation ∂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. A high order finite volume scheme on the moving mesh is used to solve this problem. Specifically, we adopt the multi-resolution weighted essentially non-oscillatory (WENO) method for the spatial discretization and a strong stability preserving (SSP) Runge-Kutta method for the temporal discretization. The remapping algorithm is high order accurate under very mild smoothness requirement (Lipschitz continuity) on the mesh movement velocity, which can always be satisfied with a suitable choice of the final pseudo-time T. Furthermore, we design our remapping algorithm to have positivity-preserving property by using the linear scaling positivity-preserving limiter so that the algorithm could ensure the positivity-preserving property of relevant physical variables and maintain conservation and original order of accuracy. A series of numerical experiments are given to demonstrate the properties of our remapping algorithm such as high order accuracy, essentially non-oscillatory performance, positivity-preserving and high computational efficiency.