Ordered upwind methods for static Hamilton-Jacobi equations.

Abstract

We introduce a family of fast ordered upwind methods for approximating solutions to a wide class of static Hamilton-Jacobi equations with Dirichlet boundary conditions. Standard techniques often rely on iteration to converge to the solution of a discretized version of the partial differential equation. Our fast methods avoid iteration through a careful use of information about the characteristic directions of the underlying partial differential equation. These techniques are of complexity O(M log M), where M is the total number of points in the domain. We consider anisotropic test problems in optimal control, seismology, and paths on surfaces.