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：本文针对进化型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条件Krylov子空间方法。核心在于通过离散正弦变换矩阵高效对角化，构建了两种新型对称正定预条件子。具体而言，我们首先对原始线性系统进行置换以得到对称系统，随后基于修正矩阵的谱符号发展出所需预条件子。理论分析表明：预条件矩阵序列的特征值聚集在±1附近，使得最小残差法能够快速收敛；若采用法方程上的共轭梯度法，预条件矩阵序列的特征值则聚集在1附近。对于高阶后向差分时间离散格式，我们给出了所提预条件方法的扩展方案，可适用于广泛的时间依赖型方程。数值算例（含变系数情形）验证了所提预条件子的有效性，其持续优于相关文献中讨论的现有块循环预条件子。