Non-oscillatory methods for relaxation approximation of Hamilton-Jacobi equations

In this paper, a class of high order non-oscillatory methods based on relaxation approximation for solving Hamilton-Jacobi equations is presented. The relaxation approximation transforms the nonlinear weakly hyperbolic equations to a semilinear strongly hyperbolic system with linear characteristic speeds and stiff source terms. The main ideas are to apply the weighted essentially non-oscillatory (WENO) reconstruction for the spatial discretization and an implicit-explicit method for the temporal integration. To illustrate the performance of the method, numerical results are carried out on several test problems for the two-dimensional Hamilton-Jacobi equations with both convex and nonconvex Hamiltonians.