A rapid numerical method for the Mullins-Sekerka flow with application to contact angle problems

TL;DR

This paper introduces a fast numerical method using charge simulation for solving the Mullins-Sekerka flow in 2D, including contact angle conditions, with new benchmarks and convergence criteria.

Contribution

It extends existing schemes by incorporating contact angle conditions and proposes a new benchmark function for validation.

Findings

The scheme effectively models Mullins-Sekerka flow with contact angles.

A sufficient condition for collocation points ensures decreasing polygonal length.

The new benchmark confirms the scheme's accuracy and stability.

Abstract

The Mullins-Sekerka problem is numerically solved in $\mathbb{R}^2$ with the aid of the charge simulation method. This is an expansion of the numerical scheme by which Sakakibara and Yazaki computed the Hele-Shaw flow. We investigate a sufficient condition for the number of collocation points to ensure that the length of the generated approximate polygonal curves gradually decreases. We propose a new benchmark function for the Mullins-Sekerka flow to confirm that the scheme works well. Moreover, by changing the fundamental solutions of the charge simulation method, we are successful to establish a numerical scheme that can be used to treat the Mullins-Sekerka problem with the contact angle condition.