An embedded method-of-lines approach to solving partial differential equations on surfaces

TL;DR

This paper presents a novel embedded method-of-lines approach for solving PDEs on surfaces using the closest point method, enabling standard numerical schemes to be applied efficiently while ensuring consistency with surface equations.

Contribution

It introduces a new formulation that generalizes existing closest point methods, allowing for straightforward convergence analysis and standard time-stepping schemes.

Findings

Method is consistent with surface PDE solutions

Compatible with implicit and explicit time-stepping

Generalizes existing closest point formulations

Abstract

We introduce a method-of-lines formulation of the closest point method, a numerical technique for solving partial differential equations (PDEs) defined on surfaces. This is an embedding method, which uses an implicit representation of the surface in a band containing the surface. We define a modified equation in the band, obtained in a straightforward way from the original evolution PDE, and show that the solutions of this equation are consistent with those of the surface equation. The resulting system can then be solved with standard implicit or explicit time-stepping schemes, and the solutions in the band can be restricted to the surface. Our derivation generalizes existing formulations of the closest point method and is amenable to standard convergence analysis.