Direct discontinuous Galerkin method for nonlinear reaction-diffusion systems in pattern formation

Nonlinear reaction-diffusion systems are often employed in mathematical modeling for pattern formation. Most of the work to date has been concerned within one-dimensional or rectangular domains. However, it is recognised that in most applications multidimensional complex geometrical domains are typically more important. In this paper we solve reaction-diffusion systems by combining direct discontinuous Galerkin (DDG) finite element methods with implicit integration factor (IIF) time integration method, on triangular meshes. This allows us solve the nonlinear algebraic systems on an element-by-element bases with significant gains in computational time. Numerical solutions of two reaction-diffusion systems, the well-studied Schnakenberg model and chloride-iodide-malonic acid (CIMA) reactive model, are presented to demonstrate effects of various domain geometries on the resulting biological patterns. Our numerical results are in good agreement with other numerical and analytical results, and with experimental results.