Abstract: Based on a simple projection of the solution increments of the underlying partial differential equations (PDE) 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 laws (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 $L^p$-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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