A novel fourth-order finite difference formula coupling the Crank-Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction-diffusion equations in two dimensions. Theoretically, under the Lipschitz assumption on the nonlinear term, the proposed high-order scheme is proved to be unconditionally stable and convergent in the discrete $L_2$-norm. Moreover, a $\tau$-matrix based preconditioner is developed to speed up the convergence of the conjugate gradient method with an optimal convergence rate (a convergence rate independent of mesh sizes) for solving the symmetric discrete linear system. Theoretical analysis shows that the spectra of the preconditioned matrices are uniformly bounded in the open interval $(3/8,2)$. To the best of our knowledge, this is the first attempt to develop a preconditioned iterative solver with a mesh-independent convergence rate for the linearized high-order scheme. Numerical examples are given to validate the accuracy of the scheme and the effectiveness of the proposed preconditioned solver.

翻译：本文提出了一种耦合Crank-Nicolson显式线性化方法的新型四阶有限差分格式，用于求解二维Riesz空间分数阶非线性反应-扩散方程。理论上，在非线性项满足Lipschitz假设条件下，证明了该高阶格式在离散$L_2$范数下是无条件稳定且收敛的。此外，发展了一种基于$\tau$矩阵的预处理子，用于加速求解对称离散线性系统的共轭梯度法，并实现了最优收敛率（与网格尺寸无关的收敛率）。理论分析表明，预处理矩阵的谱一致地包含在开区间$(3/8,2)$内。据我们所知，这是首次针对线性化高阶格式发展具有网格无关收敛率的预处理迭代求解器。数值算例验证了该格式的精度以及所提预处理求解器的有效性。