A Structure-Preserving Method of Fundamental Solutions for the Multi-Phase Mullins-Sekerka Flow

Abstract

A charge simulation method is applied to approximate the multi-phase Mullins--Sekerka flow in ${\mathbb{R}}^2$ and in a half-plane bounded by a Neumann wall. In the underlying mathematical model, interfaces driven by their curvature are coupled through a harmonic chemical-potential field. We use a charge simulation method, a variant of the method of fundamental solutions: each chemical potential is represented by fundamental solutions centered at charge points off the curve, so no bulk mesh is required. It treats curve networks separating several phases at triple junctions, including phases that occupy more than one region; on the half-plane boundary, the no-flux condition is imposed exactly by image charges, and mobile contacts stay orthogonal to the wall. The discretization is structure-preserving in the sense that every bounded phase area is conserved to machine precision at the velocity level by a null-space projection of the discrete area constraints. The proposed scheme is assessed through a convergence test against an exact three-concentric-circle solution.

35 pages, 12 figures, 4 tables

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