Third and Fourth Order Weighted ENO Schemes for Hamilton-Jacobi Equations on 2D Unstructured Meshes

Abstract

We present our recent work on designing third and fourth order accurate weighted essentially non-oscillatory (WENO) schemes for solving the nonlinear Hamilton-Jacobi equations on two-dimensional unstructured meshes. The main ideas are nodal based approximations, the usage of monotone Hamiltonians as building blocks on unstructured meshes, nonlinear weights using smooth indicators of second and higher derivatives, and a strategy to choose diversified smaller stencils to make up the bigger stencil in the WENO procedure. Both third-order and fourth-order WENO schemes use combinations of second-order approximations with nonlinear weights. A brief introduction to the methods and a selected few numerical experiments are included here.