Example MATLAB Code from Computing fluid–fluid interface evolution in Hele-Shaw cells using rational approximation

We deploy a rational solver to compute moving boundary problems in Hele-Shaw cells and porous media. At each time, a quasi-stationary Laplace problem for the pressure is solved using rational approximation. This furnishes the interfacial velocity from which the moving boundary is updated. Two approaches for representing the moving boundary are presented: orthonormal basis functions (such as Chebyshev polynomials) and level-set functions. For fluid domains with corners that are stationary or translate in a simple fashion, poles are exponentially clustered exterior to the fluid domain in the vicinity of the corner and moved in step with the respective vertex. In the case of more complicated flows, we use the adaptive Antoulas–Anderson (AAA) algorithm to select the location of poles at each time-based adaptively upon the interface shape and boundary conditions. The accuracy of the numerical method is verified by comparison with various exact solutions. The method has wide applicability, which we demonstrate through examples including porous gravity currents, one- and two-phase instabilities and flow in doubly connected regions. Key advantages of our method are its speed, relative simplicity of implementation, and ability to handle interfaces with corners and regions of high curvature at low computational cost.