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.

本文涉及一类二维时空分数对流扩散方程的数值计算和分析。导出了一个隐式差分方案来求解此类方程。证明了在某些合适的条件下，导出的差分格式是稳定且收敛的。此外，还给出了该方案在时间和空间上的收敛顺序。为了加快收敛速度​​，通过将Kronecker乘积分解（KPS）预处理器与广义最小残差（GMRES）方法相结合，介绍了一种实现差分方案的预处理策略。最后，通过几个数值例子说明了该方法的计算精度和效率。