In this work, we propose a simple yet generic preconditioned Krylov subspace method for a large class of nonsymmetric block Toeplitz all-at-once systems arising from discretizing evolutionary partial differential equations. Namely, our main result is to propose two novel symmetric positive definite preconditioners, which can be efficiently diagonalized by the discrete sine transform matrix. More specifically, our approach is to first permute the original linear system to obtain a symmetric one, and subsequently develop desired preconditioners based on the spectral symbol of the modified matrix. Then, we show that the eigenvalues of the preconditioned matrix sequences are clustered around $\pm 1$, which entails rapid convergence when the minimal residual method is devised. Alternatively, when the conjugate gradient method on the normal equations is used, we show that our preconditioner is effective in the sense that the eigenvalues of the preconditioned matrix sequence are clustered around unity. An extension of our proposed preconditioned method is given for high-order backward difference time discretization schemes, which can be applied on a wide range of time-dependent equations. Numerical examples are given, also in the variable-coefficient setting, to demonstrate the effectiveness of our proposed preconditioners, which consistently outperforms an existing block circulant preconditioner discussed in the relevant literature.

翻译：本文针对演化偏微分方程离散化产生的一类大规模非对称块Toeplitz全时段系统，提出了一种简单而通用的预处理Krylov子空间方法。具体而言，我们的主要结果是提出了两种新颖的对称正定预处理器，可通过离散正弦变换矩阵高效对角化。更详细地说，我们的方法是先对原始线性系统进行重排以获得对称系统，随后基于修正矩阵的谱符号构造所需的预处理器。然后，我们证明了预处理矩阵序列的特征值聚集于±1附近，这使得最小残差方法能够快速收敛。或者，当使用正规方程上的共轭梯度法时，我们证明了预处理矩阵序列的特征值聚集于1附近，从而验证了预处理器的有效性。本文还针对高阶向后差分时间离散格式扩展了所提出的预处理方法，可应用于广泛的时间相关方程。数值算例（包括变系数情形）验证了我们所提预处理器的有效性，其性能始终优于相关文献中讨论的现有块循环预处理器。