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

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.