Efficient Fourier-Like Legendre-Galerkin Methods for Multi-Dimensional Reaction-Diffusion Systems

High-accurate numerical methods are developed for solving multi-dimensional reaction-diffusion systems with non-periodic boundary conditions including Dirichlet, Neumann, Robin boundary conditions. The key to the efficiency of these methods is to carry out a fully diagonal representation of discretized diffusion term by the Legendre-Galerkin method with Fourier-like basis functions. The spatial semi-discretization leads to a system of coupled ordinary differential equations (ODEs). This system of ODEs is solved by an exponential time differencing spectral deferred correction (ETDSDC) scheme, leading to efficient algorithms which can achieve spectral accuracy in space, fourth and higher order (order up to 10 in this work) accuracy in time. We present high-fidelity numerical simulations for several two-dimensional reaction-diffusion systems, such as the Gray-Scott, Schnakenberg, FitzHugh-Nagumo equations and the labyrinthine patterns. Ample numerical experiments are provided to illustrate the effectiveness of our proposed methods.