A parallel preconditioner for the all-at-once linear system from evolutionary PDEs with Crank-Nicolson discretization

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: $\{z\in \mathbb{C}:1\mathrm{/}(1+\alpha )<\mathrm{\mid }z\mathrm{\mid }<1\mathrm{/}(1-\alpha )\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Re }\mathrm{e}(z)>0\}$, 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.