Abstract In this paper, we develop high order numerical schemes for the solution of the initial-boundary value problem of one-dimensional and two-dimensional space fractional diffusion equations of orders belonging to the interval ( 1 , 2 ) . Firstly, certain weighted and shifted Grunwald difference (WSGD) operator is used to approximate space Riemann-Liouville fractional derivatives, resulting in a linear system of ordinary differential equations (ODEs). Then an implicit Runge-Kutta (IRK) method is applied to discretize the resulted ODEs. Thus, we get an IRK-WSGD method for the fractional diffusion equation. We prove that under certain hypotheses, the proposed IRK-WSGD schemes are stable and have temporally fourth order accuracy and spatially second/third order accuracy. Preconditioning for discretization linear systems is discussed. Numerical experiments are presented to illustrate the accuracy and efficiency of the method.

中文翻译：

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空间分数扩散方程的 IRK-WSGD 方法

摘要 在本文中，我们开发了求解属于区间 (1, 2) 阶次的一维和二维空间分数扩散方程初边值问题的高阶数值格式。首先，使用一定的加权和移位 Grunwald 差分 (WSGD) 算子来逼近空间 Riemann-Liouville 分数阶导数，从而得到常微分方程 (ODE) 的线性系统。然后应用隐式 Runge-Kutta (IRK) 方法对结果 ODE 进行离散化。因此，我们得到了分数扩散方程的 IRK-WSGD 方法。我们证明，在某些假设下，所提出的 IRK-WSGD 方案是稳定的，并且具有时间上的四阶精度和空间上的二/三阶精度。讨论了离散化线性系统的预处理。