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 will be 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: $\left\{z\in\mathbb{C}: 1/(1 + \alpha) < |z| < 1/(1 - \alpha)~{\rm and}~\Re{e}(z) > 0\right\}$, where $0 < \alpha < 1$ is a free parameter. Meanwhile, the efficient implementation of this proposed preconditioner is described and a mesh-independent convergence rate of the preconditioned GMRES method is derived under certain conditions. Finally, we will verify our theoretical findings via numerical experiments on financial option pricing partial differential equations.

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）的著名时间积分方法。由于任意时刻的解依赖于前一时间步的解，CN方法本质上难以实现并行化。本文研究基于CN格式求解演化型PDEs的并行方法。通过采用全耦合策略，我们可在标准迭代法中使用可时间并行化的预条件子同时求解所有时间步。得益于所提预条件子的对角化特性，可证明预条件矩阵的大多数特征值等于1，其余特征值位于集合：$\left\{z\in\mathbb{C}: 1/(1 + \alpha) < |z| < 1/(1 - \alpha)~{\rm and}~\Re{e}(z) > 0\right\}$，其中$0 < \alpha < 1$为自由参数。同时，本文描述了该预条件子的高效实现方案，并在特定条件下推导了预条件GMRES方法的网格无关收敛速率。最后，通过金融期权定价偏微分方程的数值实验验证理论结果。