A new fifth-order symmetrical WENO-Z scheme for solving Hamilton-Jacobi equations

Abstract

This research describes a new fifth-order finite difference symmetrical WENO-Z scheme for solving Hamilton-Jacobi equations. This method employs the same six-point stencil as the original fifth-order WENO scheme (SIAM J. Sci. Comput. 21 (2000) 2126--2143) and a new WENO scheme recently proposed (Numer. Methods Partial Differential Eq. 33 (2017) 1095--1113), and could generate better results and create the same order of accuracy in smooth area without loss of accuracy at critical points simultaneously avoiding incorrect oscillations in the vicinity of the singularities. The new reconstruction is a convex combination of a fifth-order linear reconstruction and three third-order linear reconstructions. We prepare a detailed analysis of the approximation order of the designed WENO scheme. Some benchmark tests in 1D, 2D and 3D are performed to display the capability of the scheme.