Efficient computational models for pattern formation in fixed and evolving domains

Abstract

We efficiently model spatial patterns formed by nonlinear reaction-diffusion equations for benchmark reaction kinetics. Computational methods for modeling reaction-diffusion equations have been presented extensively in literature. Efficiency in these computational methods, either higher convergence or reduced computation time, is desired. We use a moving finite element method presented in literature and adapt it to include a second order convergence discretization and linearization. An algorithm is presented that utilizes these higher convergence methods. Numerical results demonstrate the order of convergence and reduced computational times required to model pattern formation on stationary and time dependent spatial domains. Mode isolation using manipulation of the Turing parameter space is conducted for validation. Pattern evolution on time dependent spatial domains is demonstrated.