Fast methods for the Eikonal and related Hamilton– Jacobi equations on unstructured meshes

The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a Dijkstra-like programming approach. In this paper, we discuss several extensions to this technique, including higher order versions on unstructured meshes in R(n) and on manifolds and connections to more general static Hamilton–Jacobi equations.