WENO schemes with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes with a Lax-Wendroff time discretization procedure, termed WENO-LW schemes, for solving Hamilton-Jacobi equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge-Kutta time discretizations. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with the original WENO with Runge-Kutta time discretizations schemes (WENO-RK) of Jiang and Peng [G. Jiang, D. Peng, Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 21 (2000) 2126-2143] for Hamilton-Jacobi equations, the major advantages of WENO-LW schemes are more cost effective for certain problems and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.