A Filtering technique for System of Reaction Diffusion equations

We present here a fast parallel solver designed for a system of reaction convection diffusion equations. Typical applications are large scale computing of air quality models or numerical simulation of population models where several colonies compete. Reaction-Diffusion systems can be integrated in time by point-wise Newton iteration when all space dependent terms are explicit in the time integration. Such methods are easy to code and have scalable parallelism, but are numerically inefficient. An alternative method is to use operator splitting, decoupling the time integration of reaction from convection-diffusion. However, such methods may not be time accurate thanks to the stiffness of the reaction term and are complex to parallelize with good scalability. A second alternative is to use matrix free Newton-Krylov methods. These techniques are particularly efficient provided that a good parallel preconditioner is customized to the application. The method is then not trivial to implement. We propose here a new family of fast, easy to code and numerically efficient reaction-diffusion solvers based on a filtering technique that stabilizes the explicit treatment of the diffusion terms. The scheme is completely explicit with respect to space, and the postprocessing to stabilize time stepping uses a simple FFT. We demonstrate the potential of this numerical scheme with two examples in air quality models and have compared our solution to classical schemes for two non linear reaction-diffusion problems. Further, we demonstrate on critical components of the algorithm the high potential of parallelism of our method on medium scale parallel computers. Index Terms PDE, time integration, filter, Fourier transform, reaction diffusion