WENO schemes with adaptive order for Hamilton–Jacobi equations

In this work, a fifth-order weighted essentially nonoscillatory scheme based on Legendre polynomials is constructed for simulating Hamilton–Jacobi (HJ) equations in a finite difference framework. The new reconstruction is a convex combination of a fourth-degree polynomial and two quadratic polynomials in WENO-Z fashion. This reconstruction uses the same six-point information as the original fifth-order WENO scheme [G.-S. Jiang and D. Peng, SIAM J. Sci. Comput. 21, 2126 (2000)] and could obtain smaller absolute truncation errors and the same accuracy order in the smooth region, while it has less computational time. A detailed analysis of the approximation order of the designed WENO scheme is prepared. Some benchmark tests in one-dimensional and multi-dimensional space are considered to display the capability of the new proposed scheme.