Exploiting ISAT to solve the reaction–diffusion equation

Abstract

We develop and demonstrate a computationally efficient numerical splitting technique for solving the reaction–diffusion equation. The scheme is based on the Strang splitting technique wherein the portions of the governing equations containing stiff chemical reaction terms are separated from those parts containing the less-stiff transport terms. As demonstrated, the scheme achieves second-order accuracy in space through the use of centred finite differences; second-order accuracy in time is achieved through Strang splitting. To improve greatly the computational efficiency, the pure reaction sub-steps use in situ adaptive tabulation (ISAT) to compute efficiently the reaction mapping while the pure diffusion sub-steps use an implicit Crank–Nicolson finite-difference method. The scheme is applied to an unsteady one-dimensional reaction–diffusion model equation with detailed chemical kinetics. For this test problem, we show spatial and temporal convergence results, the impacts of ISAT and ODE solver error tolerances, and demonstrate computational speed-ups achieved by using ISAT over direct integration.