An interpolatory directional splitting-local discontinuous Galerkin method with application to pattern formation in 2D/3D

An efficient computational method to approximate the solution of a general class of nonlinear reaction-diffusion systems in Cartesian grids is presented. The proposed scheme uses the Local Discontinuous Galerkin (LDG) method as spatial discretization and the symmetric Strang operator splitting as time marching scheme. As a result, not only diffusion and reaction are decoupled; but the reaction sub-step reduces to a set of completely independent small local nonlinear systems of ordinary differential equations. To reduce their computational cost a Lagrange interpolatory technique is used. Diffusion sub-steps are approximated by a directional splitting method derived from an algebraic decomposition of the discrete Laplacian operator, which amounts to solve sets of completely independent small block tridiagonal linear systems; one set per direction. Stability of the diffusive step is analyzed. Finally the efficiency and accuracy of the proposed scheme is numerically investigated on well known Turing pattern formation models in 2D/3D.