In [McDonald, Pestana and Wathen, \textit{SIAM J. Sci. Comput.}, 40 (2018), pp. A1012--A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial differential equations, in which the preconditioned matrix is proven to be diagonalizable and to have identity-plus-low-rank decomposition in the case of the heat equation. In this paper, we generalize the block circulant preconditioner by introducing a small parameter $\epsilon>0$ into the top-right block of the block circulant preconditioner. The implementation of the generalized preconditioner requires the same computational complexity as that of the block circulant one.Theoretically, we prove that (i) the generalization preserves the diagonalizability and the identity-plus-low-rank decomposition; (ii) all eigenvalues of the new preconditioned matrix are clustered at 1 for sufficiently small $\epsilon$; (iii) GMRES method for the preconditioned system has a linear convergence rate independent of size of the linear system when $\epsilon$ is taken to be smaller than or comparable to square root of time-step size. Numerical results are reported to confirm the efficiency of the proposed preconditioner and to show that the generalization improves the performance of block circulant preconditioner.

翻译：在[McDonald, Pestana 和 Wathen,\ textit{SIAM J. Sci. Comput.}, 40 (2018), pp. A1012-A1033], 提议对进化部分差异方程式产生的全天线线性系统设置一个块状螺旋前置先决条件,在这种系统中,前提条件矩阵被证明可以进行分解,并在热方程中保持身份-加-低级分解。在本文中,我们通过将一个小参数 $\epsilon>0$引入区块的顶端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端口端端端端端端端端端端端口端端端端端端端口端端端端端口端端端端端端端端端端端口端端端端端端端端端端端端端端端端端口端端端端端端端端端端端端端端端端端端端端端端端端口端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端,且端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端端