The random walk model describing the super-diffusion competition phenomenon of particles can derive the spatial fractional diffusion equation. For irregular diffusion and super-diffusion phenomena, the use of such equation can obtain more accurate and realistic results, so it has a wide application background in practice. The implicit finite-difference method derived from the shifted Grünwald scheme is used to discretize the spatial fractional diffusion equation. The coefficient matrix of discrete system is in the form of the sum of a diagonal matrix and a Toeplitz matrix. In this paper, a preconditioner is constructed, which transforms the coefficient matrix into the form of an identity matrix plus a diagonal matrix multiplied by Toeplitz matrix. On this basis, a new quasi-Toeplitz trigonometric transform splitting iteration format (abbreviated as QTTTS method) is proposed. We theoretically verify the unconditional convergence of the new method, and obtain the effective optimal form of the iteration parameter. Finally, numerical simulation experiments also demonstrate the accurateness and efficiency of the new method.

描述粒子超扩散竞争现象的随机游走模型可以推导出空间分数扩散方程。对于不规则扩散和超扩散现象，使用这样的方程可以获得更准确和逼真的结果，因此在实践中具有广泛的应用背景。从移位 Grünwald 方案导出的隐式有限差分方法用于离散空间分数扩散方程。离散系统的系数矩阵是对角矩阵和托普利兹矩阵之和的形式。本文构造了一个预处理器，将系数矩阵转化为单位矩阵加对角矩阵乘以托普利茨矩阵的形式。在此基础上，提出了一种新的拟托普利兹三角变换分裂迭代格式（简称QTTTS方法）。我们从理论上验证了新方法的无条件收敛性，并得到迭代参数的有效最优形式。最后，数值模拟实验也证明了新方法的准确性和有效性。