In this paper, we consider diagonal-times-nonsymmetric-Toeplitz (DNT) preconditioner for Toeplitz-like linear systems arising from one-sided space fractional diffusion equations (OSFDE) with variable coefficients. We first correct the result of Lemma 3.3 in Lin et al. (BIT Numer Math 58, 729–748, 2018) and prove the corrected result. We then extend the DNT preconditioner to Toeplitz-like linear systems arising from OSFDEs, where high order difference operators are applied to discretize the fractional derivative. Theoretically, we prove that the condition number of the preconditioned matrix is uniformly bounded by a constant under mild assumptions and verify that several discretization schemes from the literature satisfy the required assumptions. Numerical results are reported to show the efficiency of the DNT preconditioner.

在本文中，我们考虑了由变系数的单侧空间分数阶扩散方程（OSFDE）产生的类Toeplitz线性系统的对角时间非对称Toeplitz（DNT）预处理器。我们首先在Lin等人中校正引理3.3的结果。（BIT Numer Math 58，729–748，2018）并证明更正后的结果。然后，我们将DNT预调节器扩展到OSFDE产生的Toeplitz线性系统，在其中应用高阶差分算子离散化分数导数。从理论上讲，我们证明了预处理矩阵的条件数在一个温和的假设下均匀地由一个常数界定，并验证了文献中的几种离散化方案都满足了所需的假设。数值结果报道表明，DNT预处理器的效率。