An Embedding Technique for the Solution of Reaction-Diffusion Equations on Algebraic Surfaces with Isolated Singularities

TL;DR

This paper introduces a novel embedding method for solving reaction-diffusion equations on algebraic curves with isolated singularities by desingularization, surface approximation, and the Closest Point Method, enabling numerical solutions on complex geometries.

Contribution

It presents a desingularization-based embedding technique combined with the Closest Point Method for reaction-diffusion PDEs on algebraic surfaces with singularities, extending applicability to higher dimensions.

Findings

The method accurately approximates solutions on singular algebraic curves.

Numerical experiments demonstrate the effectiveness of the approach.

Potential for generalization to higher-dimensional surfaces with multiple singularities.

Abstract

In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities.