Convergence Rate for the Ordered Upwind Method

TL;DR

This paper proves that the Ordered Upwind Method converges at a rate proportional to the square root of mesh size when approximating solutions to the Hamilton-Jacobi-Bellman equation, providing theoretical convergence guarantees.

Contribution

It establishes the first theoretical convergence rate for the Ordered Upwind Method in solving static HJB equations on unstructured meshes.

Findings

Convergence rate is at least the square root of the mesh size.

Numerical examples confirm the theoretical convergence rate.

Provides a theoretical foundation for the accuracy of OUM.

Abstract

The Ordered Upwind Method (OUM) is used to approximate the viscosity solution of the static Hamilton-Jacobi-Bellman (HJB) with direction-dependent weights on unstructured meshes. The method has been previously shown to provide a solution that converges to the exact solution, but no convergence rate has been theoretically proven. In this paper, it is shown that the solutions produced by the OUM in the boundary value formulation converge at a rate of at least the square root of the largest edge length in the mesh in terms of maximum error. An example with similar order of numerical convergence is provided.