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

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Abstract

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.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Science, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama-shi, Okayama, 700-0005, Japan

    Koya Sakakibara

  2. RIKEN iTHEMS, 2-1 Hirosawa, Wako-shi, Saitama, 351-0198, Japan

    Koya Sakakibara

  3. Graduate School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa, 214-8571, Japan

    Yusaku Shimoji

  4. Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa, 214-8571, Japan

    Shigetoshi Yazaki

Authors
  1. Koya Sakakibara
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  2. Yusaku Shimoji
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  3. Shigetoshi Yazaki
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Correspondence to Yusaku Shimoji.

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Sakakibara, K., Shimoji, Y. & Yazaki, S. A simple numerical method for Hele–Shaw type problems by the method of fundamental solutions. Japan J. Indust. Appl. Math. 39, 869–887 (2022). https://doi.org/10.1007/s13160-022-00530-1

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  • DOI: https://doi.org/10.1007/s13160-022-00530-1

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