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

Abstract

We derive a Godunov-type numerical flux for the class of strictly convex, homogeneous Hamiltonians that includes H(p, q) = ap 2 + bq 2 − 2cpq, c 2 < ab. We combine our Godunov numerical fluxes with simple GaussSeidel type iterations for solving the corresponding Hamilton-Jacobi 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 provides a 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.

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

“…Min and Gibou also used the idea of Russo and Smereka with slight modifications in the context of adaptive mesh refinement [90], and Min pointed out that it is advantageous in terms of speed and memory to replace the traditional Runge-Kutta scheme in time with a Gauss-Seidel iteration of the forward Euler scheme [88]. Finally, we mention that other techniques can be used to reinitialize φ as a distance function [125,124,149,166,148,31,147,53], each with their pros and cons. We refer the interested readers to the book by Osher and Fedkiw [101] as well to the book by Sethian [127] for more details on the level-set method.…”

A procedure for calculating the many-particle Bohm quantum potential

“…This procedure, by making a fully symmetric [25,26,27,28,29,30,31,32,33] and references therein). In particular the fast sweeping algorithms of Ref.…”

“…In particular the fast sweeping algorithms of Ref. [30], and the robust algorithms for multidimensional Hamilton-Jacobi equations of Refs. [31,32,33] represent an extremely useful approach to a computer implementation of this procedure.…”

Filtered schemes for Hamilton–Jacobi equations: A simple construction of convergent accurate difference schemes

“…The schemes are formally accurate: we implement second, third and fourth order accurate schemes in one dimension and second order accurate schemes in two dimensions, indicating how to build higher order ones. They are also explicit, which means they can be solved using the fast sweeping method [TCOZ03,Zha05], or the fast marching method [Set99,Tsi95] in the case of the eikonal equation.…”