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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.
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