An adaptive mesh redistribution method for nonlinear Hamilton–Jacobi equations in two- and three-dimensions

Abstract

This paper presents an adaptive mesh redistribution (AMR) method for solving the nonlinear Hamilton–Jacobi equations and level-set equations in two- and three-dimensions. Our approach includes two key ingredients: a non-conservative second-order interpolation on the updated adaptive grids, and a class of monitor functions (or indicators) suitable for the Hamilton–Jacobi problems. The proposed adaptive mesh methods transform a uniform mesh in the logical domain to cluster grid points at the regions of the physical domain where the solution or its derivative is singular or nearly singular. Moreover, the formal second-order rate of convergence is preserved for the proposed AMR methods. Extensive numerical experiments are performed to demonstrate the efficiency and robustness of the proposed adaptive mesh algorithm.