In this paper, high-order finite difference methods are proposed to solve the initial-boundary value problem for space Riesz variable-order fractional diffusion equations. Based on weighted-shifted-Grünwald-difference (WSGD) operators proposed in Lin and Liu (J. Comput. Appl. Math. 363, 77–91 (2020)) for Riemann-Liouville fractional derivatives, we derive WSGD operators for variable-order ones by using the relation between variable-order fractional derivative and (constant-order) fractional derivative. We then apply Crank-Nicolson-weighted-shifted-Grünwald-difference (CN-WSGD) schemes to the initial-boundary problem for space Riesz variable-order diffusion equations. Theoretical results on the stability and convergence of CN-WSGD schemes are presented and proved. Moreover, we derive a problem-based method to choose suitable CN-WSGD schemes, which leads to unconditioned stable linear systems with optimal upper bound for accuracy. Numerical results show that the proposed schemes are very efficient.

本文提出了高阶有限差分方法来解决空间Riesz变阶分数阶扩散方程的初边值问题。基于加权移-Grünwald的差（WSGD）运营商在Lin和刘（J. COMPUT提出申请。数学。363（77-91（2020）），对于黎曼-利维尔分数阶导数，我们利用变量阶数导数和（常数阶）分数导数之间的关系推导了WSGD算子。然后，我们将Crank-Nicolson加权移位的Grünwald差分（CN-WSGD）方案应用于空间Riesz变阶扩散方程的初边界问题。提出并证明了CN-WSGD方案的稳定性和收敛性的理论结果。此外，我们推导了一种基于问题的方法来选择合适的CN-WSGD方案，这会导致无条件的稳定线性系统的精度达到最佳上限。数值结果表明，该方案是非常有效的。