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

• Published in: Japan journal of industrial and applied mathematics Vol. 39; no. 3; pp. 869 - 887
• Main Authors: Sakakibara, Koya, Shimoji, Yusaku, Yazaki, Shigetoshi
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01-12-2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Asymptotic methods; Asymptotic properties; Computational Mathematics and Numerical Analysis; Magnetic fluids; Mathematics; Mathematics and Statistics; Meshless methods; Numerical methods; Original Paper; 65N35; 65N80; Hele–Shaw flow; Amano’s method; 76E25; 76D27; 35J05; Magnetic fluid; Volume-preserving property; The method of fundamental solutions; Time-dependent gap
• ISSN: 0916-7005; 1868-937X
• DOI: 10.1007/s13160-022-00530-1

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, the numerical method satisfies the volume-preserving property combining with the asymptotic uniform distribution method. 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.