Corner cases, singularities, and dynamic factoring

TL;DR

This paper introduces a dynamic factoring method for Eikonal equations that identifies and addresses rarefaction singularities caused by boundary and coefficient discontinuities, improving convergence in complex 2D scenarios.

Contribution

It extends factoring techniques to handle unknown, dynamically detected rarefactions in 2D Eikonal equations, generalizing the Fast Marching Method.

Findings

Restores first-order convergence in complex scenarios

Effectively handles boundary-induced rarefactions

Demonstrates improved maze navigation performance

Abstract

In Eikonal equations, rarefaction is a common phenomenon known to degrade the rate of convergence of numerical methods. The `factoring' approach alleviates this difficulty by deriving a PDE for a new (locally smooth) variable while capturing the rarefaction-related singularity in a known (non-smooth) `factor'. Previously this technique was successfully used to address rarefaction fans arising at point sources. In this paper we show how similar ideas can be used to factor the 2D rarefactions arising due to nonsmoothness of domain boundaries or discontinuities in PDE coefficients. Locations and orientations of such rarefaction fans are not known in advance and we construct a `just-in-time factoring' method that identifies them dynamically. The resulting algorithm is a generalization of the Fast Marching Method originally introduced for the regular (unfactored) Eikonal equations. We show that our approach restores the first-order convergence and illustrate it using a range of maze navigation examples with non-permeable and `slowly permeable' obstacles.