Hamilton--Jacobi equations on an evolving surface

TL;DR

This paper extends the theory and numerical methods for Hamilton--Jacobi equations to evolving surfaces, establishing well-posedness, stability, and error bounds for the proposed finite volume scheme.

Contribution

It introduces a natural extension of viscosity solutions to evolving hypersurfaces and develops a stable, convergent numerical scheme without the need for acute triangles.

Findings

The scheme is stable and consistent.

Convergence is proven with an existence proof.

An error bound similar to stationary cases is established.

Abstract

We consider the well-posedness and numerical approximation of a Hamilton--Jacobi equation on an evolving hypersurface in $\mathbb R^3$. Definitions of viscosity sub- and supersolutions are extended in a natural way to evolving hypersurfaces and provide uniqueness by comparison. An explicit in time monotone numerical approximation is derived on evolving interpolating triangulated surfaces. The scheme relies on a finite volume discretisation which does not require acute triangles. The scheme is shown to be stable and consistent leading to an existence proof via the proof of convergence. Finally an error bound is proved of the same order as in the flat stationary case.