A Class of High Resolution Difference Schemes for Nonlinear Hamilton-Jacobi Equations with Varying Time and Space Grids

Based on a simple projection of the solution increments of the underlying partial differential equations (PDEs) at each local time level, this paper presents a difference scheme for nonlinear Hamilton--Jacobi (H--J) equations with varying time and space grids. The scheme is of good consistency and monotone under a local CFL-type condition. Moreover, one may deduce a conservative local time step scheme similar to Osher and Sanders scheme approximating hyperbolic conservation law (CL) from our scheme according to the close relation between CLs and H--J equations. Second order accurate schemes are constructed by combining the reconstruction technique with a second order accurate Runge--Kutta time discretization scheme or a Lax--Wendroff type method. They keep some good properties of the global time step schemes, including stability and convergence, and can be applied to solve numerically the initial-boundary-value problems of viscous H--J equations. They are also suitable to parallel computing. Numerical errors and the experimental rate of convergence in the Lp-norm, p = 1, 2, and $\infty$, are obtained for several one- and two-dimensional problems. The results show that the present schemes are of higher order accuracy.

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