The Crank-Nicolson (CN) method is a well-known time integrator for evolutionary partial differential equations (PDEs) arising in many real-world applications. Since the solution at any time depends on the solution at previous time steps, the CN method is inherently difficult to parallelize. In this paper, we consider a parallel method for the solution of evolutionary PDEs with the CN scheme. Using an all-at-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can prove that most eigenvalues of preconditioned matrices are equal to 1 and the others lie in the set: $\left\{z\in\mathbb{C}: 1/(1 + \alpha) < |z| < 1/(1 - \alpha)~{\rm and}~\Re{\rm e}(z) > 0\right\}$, where $0 < \alpha < 1$ is a free parameter. Besides, the efficient implementation of the proposed preconditioner is described. Given certain conditions, we prove that the preconditioned GMRES method exhibits a mesh-independent convergence rate. Finally, we will verify both theoretical findings and the efficacy of the proposed preconditioner via numerical experiments on financial option pricing PDEs.

翻译：Crank-Nicolson（CN）方法是求解众多实际应用中演化型偏微分方程（PDEs）的经典时间积分算法。由于任意时刻的解依赖于前序时刻的解，CN方法本质难以并行化。本文针对采用CN格式的演化型PDEs求解问题，提出一种并行方法。通过全时段求解策略，我们可在标准迭代框架内利用可并行化的时间方向预处理子同时求解所有时间步。基于所提预处理子的对角化特性，可证明预处理矩阵的绝大多数特征值等于1，其余特征值位于集合：$\left\{z\in\mathbb{C}: 1/(1 + \alpha) < |z| < 1/(1 - \alpha)~{\rm and}~\Re{\rm e}(z) > 0\right\}$中，其中$0 < \alpha < 1$为自由参数。此外，本文给出了预处理子的高效实现方案。在特定条件下，我们证明了预处理GMRES方法具有网格无关收敛速度。最后，通过金融期权定价PDEs的数值实验，验证了理论发现与所提预处理子的有效性。