WENO-Z scheme with new nonlinear weights for Hamilton-Jacobi equations and adaptive approximation

A new fifth-order weighted essentially nonoscillatory (WENO) scheme is designed to approximate Hamilton-Jacobi equations. As employing a fifth-order linear approximation and three third-order ones on the same six-point stencil as before, a newly considered WENO-Z methodology is adapted to define nonlinear weights and the final WENO reconstruction results in a simple and clear convex combination. The scheme has formal fifth-order accuracy in smooth regions of the solution and nonoscillating behavior nearby singularities. A full account is given of the key role of parameters in WENO reconstruction and their selection. The latter half describes the adaptive stage on WENO approximation in convergence framework, which enables us to get the numerical solution to converge still achieving high-order accuracy for the nonconvex problems where the pure WENO scheme fails to converge. Detailed numerical experiments are performed to demonstrate the ability of the proposed numerical methods.