Numerical Methods for Fractional Diffusion-Reaction Problems Based on Operator Splitting and BURA

A system of time dependent fractional-in-space diffusion-reaction equations is considered. The spectral definition of fractional power of the diffusion operators is assumed. The system is coupled trough the reaction operators. The operator splitting is one of the most powerful methods to solve such problems. In this paper, we analyze a composite algorithm which integrates the sequential splitting. The convergence of the splitting scheme is discussed. The diffusion and reaction sub-processes have substantially different properties. Thus, the splitting scheme provides opportunities to utilize the best available solvers for each of the sub-tasks. The interaction of the operator splitting and the applied discretization of the sub-processes is studied. One principle novelty of the presented results concerns the treatment of the diffusion sub-tasks. The BURA (Best Uniform Rational Approximation) method is applied for numerical solution of the involved fractional-in-space diffusion problems. Sufficient conditions for balancing the different errors of the composite algorithm are derived.