Structure‐preserving numerical scheme for the one‐phase Hele‐Shaw problems by the method of fundamental solutions

Abstract

In this study, the solutions to the one‐phase interior or the classical Hele‐Shaw problem are discretized in space by employing the method of fundamental solutions combined with the uniform distribution method; furthermore, a system of ordinary differential equations is obtained, which is solved using the usual fourth‐order Runge‐Kutta method. The one‐phase interior Hele‐Shaw problem has curve‐shortening (CS), area‐preserving (AP), and barycenter‐fixed (BF) properties. Under our numerical scheme, a discrete version of CS and BF properties holds in an asymptotic sense and AP property in an exact sense, whereas in general, simple boundary element method does not satisfy these properties. Moreover, the one‐phase exterior Hele‐Shaw problem and one‐phase interior Hele‐Shaw problem containing sink/source points can be treated. Therefore, in each problem, a nontrivial exact solution is constructed and an experimental order of convergence is shown.

A fully discrete curve-shortening polygonal evolution law for moving boundary problems

“…Typical examples include the mean curvature flow [34], areapreserving mean curvature flow [10], and Hele-Shaw flow [20]. We note that it was proved in [11,10,12,14] that the first two examples do not allow any topological changes during the evolution, and numerical observations suggested that the same property holds for the Hele-Shaw flow [38].…”

“…In this section, we discretize the moving boundary problem (1.1) in space. The discretization is based on [38,41]. Below we give definitions of discrete tangent vectors, discrete outward normal vectors, discrete curvatures, discrete tangential velocities and discrete normal velocities.…”

“…The relation for the length L[Γ t ] is an immediate consequence of the definition of the discrete curvature (3.3). The relation for the enclosed area A[Ω t ] first appeared in [38,Proposition 4] without proof; therefore, we give its proof. The enclosed area A[Ω] can be expressed as…”

Method of fundamental solutions with weighted average condition and dummy points

“…where f is a given data. The invariant scheme of the method of fundamental solutions (MFS) with weighted average condition and dummy points offers an approximate solution for (1a)-(1b) as in the following procedure [1,2].…”

“…Therefore, we can add one more condition such as the weighted average condition (3), which is nothing but the invariant condition (6) when all weights H j are equal to 1. Our modifications (3)-(4) enable us to construct some geometrical structure-preserving numerical scheme for the one-phase Hele-Shaw problem [1,2], which is a moving boundary problem.…”