Hybrid Finite Difference Fifth-Order Multi-Resolution WENO Scheme for Hamilton-Jacobi Equations

Abstract

In this paper, a fifth-order hybrid multi-resolution weighted essentially non-oscillatory (WENO) scheme in the finite difference framework is proposed for solving one- and two-dimensional Hamilton-Jacobi equations. Firstly, a new discontinuity sensor is designed based on the extreme values of the highest degree polynomial in the multi-resolution WENO procedures. This hybrid strategy does not contain any human parameters related to specific problems and can identify the troubled grid points accurately and automatically. Secondly, a hybrid multi-resolution WENO scheme for Hamilton-Jacobi equations is developed based on the above discontinuity sensor and a simplified multi-resolution WENO scheme, which yields uniform high-order accuracy in smooth regions and sharply resolves discontinuities. Compared with the existing multi-resolution WENO scheme, the method developed in this paper can inherit its many advantages and is more efficient. Finally, some benchmark numerical experiments are performed to demonstrate the performance of the presented fifth-order hybrid multi-resolution WENO scheme for one- and two-dimensional Hamilton-Jacobi equations.