PDEs on moving surfaces via the closest point method and a modified grid based particle method

TL;DR

This paper introduces a novel approach combining the closest point method with a modified grid-based particle method to efficiently solve PDEs on moving surfaces, demonstrating good convergence and applicability to coupled advection-diffusion problems.

Contribution

It develops a new method integrating CPM and a modified GBPM for PDEs on moving surfaces, addressing limitations of static surface methods.

Findings

The combined method achieves numerical convergence.

Effective for advection-diffusion equations on moving surfaces.

Demonstrates applicability to strongly coupled PDEs.

Abstract

Partial differential equations (PDEs) on surfaces arise in a wide range of applications. The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is a recent embedding method that has been used to solve a variety of PDEs on smooth surfaces using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. The original closest point method (CPM) was designed for problems posed on static surfaces, however the solution of PDEs on moving surfaces is of considerable interest as well. Here we propose solving PDEs on moving surfaces using a combination of the CPM and a modification of the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993{3024, [2009]). The grid based particle method (GBPM) represents and tracks surfaces using meshless particles and an Eulerian reference grid. Our modification of the GBPM introduces a reconstruction step into the original method to ensure that all the grid points within a computational tube surrounding the surface are active. We present a number of examples to illustrate the numerical convergence properties of our combined method. Experiments for advection-diffusion equations that are strongly coupled to the velocity of the surface are also presented.