High Order Arbitrary Lagrangian-Eulerian Finite Difference WENO Scheme for Hamilton-Jacobi Equations

Abstract

In this paper, a high order arbitrary Lagrangian-Eulerian (ALE) finite difference weighted essentially non-oscillatory (WENO) method for Hamilton-Jacobi equations is developed. This method is based on moving quadrilateral meshes, which are often used in Lagrangian type methods. The algorithm is formed in two parts: spatial discretization and temporal discretization. In the spatial discretization, we choose a new type of multi-resolution WENO schemes on a nonuniform moving mesh. In the temporal discretization, we use a strong stability preserving (SSP) Runge-Kutta method on a moving mesh for which each grid point moves independently, with guaranteed high order accuracy under very mild smoothness requirement (Lipschitz continuity) for the mesh movements. Extensive numerical tests in one and two dimensions are given to demonstrate the flexibility and efficiency of our moving mesh scheme in solving both smooth problems and problems with corner singularities.