An implicit difference scheme with the KPS preconditioner for two-dimensional time-space fractional convection-diffusion equations

This paper deals with the numerical computation and analysis for a class of two-dimensional time-space fractional convection-diffusion equations. An implicit difference scheme is derived for solving this class of equations. It is proved under some suitable conditions that the derived difference scheme is stable and convergent. Moreover, the convergence orders of the scheme in time and space are also given. In order to accelerate the convergence rate, by combining the Kronecker product splitting (KPS) preconditioner with the generalized minimal residual (GMRES) method, a preconditioning strategy for implementing the difference scheme is introduced. Finally, several numerical examples are presented to illustrate the computational accuracy and efficiency of the methods.