Structure‐preserving numerical scheme for the one‐phase Hele‐Shaw problems by the method of fundamental solutions. Issue 6 (5th September 2019) - Sakakibara, Koya; Yazaki, Shigetoshi
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.