Fast Sweeping Algorithms for a Class of Hamilton--Jacobi Equations

We derive a Godunov-type numerical flux for the class of strictly convex, homogeneous Hamiltonians that includes H(p,q)=√ap²+bq²-2cpq, c²<ab. We combine our Godunov numerical fluxes with simple Gauss--Seidel-type iterations for solving the corresponding Hamilton--Jacobi (HJ) equations. The resulting algorithm is fast since it does not require a sorting strategy as found, e.g., in the fast marching method. In addition, it providesa way to compute solutions to a class of HJ equations for which the conventional fast marching method is not applicable. Our experiments indicate convergence after a few iterations, even in rather difficult cases.