Fast IIF–WENO Method on Non-uniform Meshes for Nonlinear Space-Fractional Convection–Diffusion–Reaction Equations

In this article, our goal is to establish fast and efficient numerical methods for nonlinear space-fractional convection–diffusion–reaction (CDR) equations in the 1, 2, and 3 dimensions. For the spatial discretization of the CDR equations, the weighted essentially non-oscillatory (WENO) scheme is used to approximate the convection term, and the fractional centered difference formula is applied to deal with the diffusion term. As a result, a nonlinear system of ordinary differential equations (ODEs) is derived. Since implicit integration factor (IIF) methods are a class of time-stepping schemes with good robustness and stability, the second order IIF–WENO scheme on non-uniform meshes in Jiang and Zhang (J Comput Phys 253:368–388, 2013) is applied to solve the nonlinear ODEs system. In order to obtain an efficient implementation of the IIF–WENO scheme, we propose an adaptive restarting Krylov subspace method to compute the action of matrix exponentials arising in IIF–WENO. Numerical examples are presented to confirm the validity of the IIF–WENO scheme, and to verify that the proposed fast solution algorithm is extremely attractive in terms of computational complexity and memory storage.