An IRK‐QCD Scheme for the Space Fractional Diffusion Equation and Block‐Lower‐Triangle Preconditioners for the Corresponding Linear Systems

ABSTRACT In this paper, a novel high‐order finite difference scheme with temporally and spatially fourth‐order accuracy, called the implicit Runge‐Kutta‐quasi‐compact difference (IRK‐QCD) scheme, is proposed for one‐ and two‐dimensional space fractional diffusion equations (SFDEs). Firstly, the quasi‐compact difference (QCD) operator is used to approximate the space Riemann‐Liouville fractional derivative, which results in a system of ordinary differential equations (ODEs). Then an implicit Runge‐Kutta (IRK) method is applied to discretize the resulted ODEs. The unconditionally stability and the fourth order convergence rate of the IRK‐QCD scheme are theoretically established. Block‐lower‐triangular (BLT) preconditioners are employed for the discretization linear systems. We prove that the eigenvalues of the preconditioned matrix lie in disc , independent of discretization step‐sizes and fractional order(s). We also prove that the condition number of the preconditioned matrix has a uniform upper bound. Numerical experiments, including an empirical example, are implemented to illustrate the accuracy and efficiency of the proposed method.