A numerical framework for singular limits of a class of reaction diffusion problems

We present a numerical framework for solving localized pattern structures of reaction–diffusion type far from the Turing regime. We exploit asymptotic structure in a set of well established pattern formation problems to analyze a singular limit model that avoids time and space adaptation typically associated to full numerical simulations of the same problems. The singular model involves the motion of a curve on which one of the chemical species is concentrated. The curve motion is non-local with an integral equation that has a logarithmic singularity. We generalize our scheme for various reaction terms and show its robustness to other models with logarithmic singularity structures. One such model is the 2D Mullins–Sekerka flow which we implement as a test case of the method. We then analyze a specific model problem, the saturated Gierer–Meinhardt problem, where we demonstrate dynamic patterns for a variety of parameters and curve geometries.