Local-Structure-Preserving Discontinuous Galerkin Methods with Lax-Wendroff Type Time Discretizations for Hamilton-Jacobi Equations

In this paper, a family of high order numerical methods are designed to solve the Hamilton-Jacobi equation for the viscosity solution. In particular, the methods start with a hyperbolic conservation law system closely related to the Hamilton-Jacobi equation. The compact one-step one-stage Lax-Wendroff type time discretization is then applied together with the local-structure-preserving discontinuous Galerkin spatial discretization. The resulting methods have lower computational complexity and memory usage on both structured and unstructured meshes compared with some standard numerical methods, while they are capable of capturing the viscosity solutions of Hamilton-Jacobi equations accurately and reliably. A collection of numerical experiments is presented to illustrate the performance of the methods.