A simple numerical method for Hele-Shaw type problems by the method of fundamental solutions

TL;DR

This paper introduces a simple, mesh-free numerical method based on fundamental solutions for solving Hele-Shaw problems, effectively capturing complex flow patterns while preserving volume.

Contribution

It develops a novel combination of the fundamental solutions method with asymptotic uniform distribution to accurately simulate Hele-Shaw flows with volume preservation.

Findings

High accuracy in potential problem solutions

Effective simulation of fingering patterns

Volume-preserving numerical scheme

Abstract

Hele-Shaw flows with time-dependent gaps create fingering patterns, and magnetic fluids in Hele-Shaw cells create intriguing patterns.We propose a simple numerical method for Hele-Shaw type problems by the method of fundamental solutions.The method of fundamental solutions is one of the mesh-free numerical solvers for potential problems, which provides a highly accurate approximate solution despite its simplicity.Moreover, combining with the asymptotic uniform distribution method, the numerical method satisfies the volume-preserving property.We use Amano's method to arrange the singular points in the method of fundamental solutions.We show several numerical results to exemplify the effectiveness of our numerical scheme.