Alternating Evolution Schemes for Hamilton-Jacobi Equations

In this work, we propose a high-resolution alternating evolution (AE) scheme to solve Hamilton–Jacobi equations. The construction of the AE scheme is based on an alternating evolution system of the Hamilton–Jacobi equation, following the idea previously developed for hyperbolic conservation laws. A semidiscrete scheme derives directly from a sampling of this system on alternating grids. Higher order accuracy is achieved by a combination of high order nonoscillatory polynomial reconstruction from the obtained grid values and a time discretization with matching accuracy. Local AE schemes are made possible by choosing the scale parameter to reflect the local distribution of waves. The AE schemes have the advantage of easy formulation and implementation and efficient computation of the solution. For the first local AE scheme and the second order local AE scheme with a limiter, we prove the numerical stability in the sense of satisfying the maximum principle. Numerical experiments for a set of Hamilton–Jacobi equations are presented to demonstrate both accuracy and capacity of these AE schemes.