High-fidelity simulations for Turing pattern formation in multi-dimensional Gray-Scott reaction-diffusion system

In this study, the authors present high-fidelity numerical simulations for capturing the Turing pattern in a multi-dimensional Gray-Scott reaction-diffusion system. For this purpose, an explicit mixed modal discontinuous Galerkin (DG) scheme based on multi-dimensional structured meshes is employed. This numerical scheme deals with high-order derivatives in diffusion, and highly nonlinear functions in reaction terms. Spatial discretization is accomplished using hierarchical basis functions based on scaled Legendre polynomials. A new reaction term treatment is also detailed, showing an intrinsic property of the DG scheme and preventing incorrect solutions due to excessively nonlinear reaction terms. The system is reduced to a set of time-dependent ordinary differential equations, which are solved with an explicit third-order Total Variation Diminishing (TVD) Runge-Kutta method. The Saddle-Node, Hopf, and Turing bifurcations' boundary conditions are also examined for the system, and subsequently, Turing space is identified. The developed numerical scheme is applied to various multidimensional Gray-Scott systems to assess its ability to capture Turing pattern formation. Several test problems, such as stationary and non-stationary waves, pulse-splitting waves, self-replicating waves, and different types of spots, are chosen from the literature to demonstrate the different Turing patterns. (c) 2023 Elsevier Inc. All rights reserved.