Convergence analysis of the fast sweeping method for static convex Hamilton–Jacobi equations

Songting Luo,Hongkai Zhao

doi:10.1186/s40687-016-0083-8

Songting Luo, Hongkai Zhao

Open Access

https://doi.org/10.1186/s40687-016-0083-8

Copy DOI

Abstract

In this work, we study the convergence of an efficient iterative method, the fast sweeping method (FSM), for numerically solving static convex Hamilton–Jacobi equations. First, we show the convergence of the FSM on arbitrary meshes. Then we illustrate that the combination of a contraction property of monotone upwind schemes with proper orderings can provide fast convergence for iterative methods. We show that this mechanism produces different behavior from that for elliptic problems as the mesh is refined. An equivalence between the local solver of the FSM and the Hopf formula under linear approximation is also proved. Numerical examples are presented to verify our analysis.

Highlights

Efficient and robust iterative methods are highly desirable for solving a variety of static hyperbolic partial differential equations (PDEs) numerically
Through the contraction property we study the convergence of Gauss–Seidel iterations with proper orderings
Given a vertex C and its local mesh DhC which consists of all triangles that include C as a common vertex, the local discretization scheme first uses the PDE to find a possible value at C that satisfies both the consistency and the causality condition on each triangle and picks the minimum one among all the possible values according to the control interpretation of the viscosity solution

Summary

Background

Efficient and robust iterative methods are highly desirable for solving a variety of static hyperbolic partial differential equations (PDEs) numerically. Given a vertex C and its local mesh DhC which consists of all triangles that include C as a common vertex (see Fig. 2), the local discretization scheme first uses the PDE to find a possible value at C that satisfies both the consistency and the causality condition on each triangle and picks the minimum one among all the possible values according to the control interpretation (or the Hopf formula) of the viscosity solution. With the above properties for the discretization scheme, we prove the convergence of the iterative method and the convergence of the numerical solution to the viscosity solution as the grid size approaches zero.

Local truncation error and error estimate

Conclusion