2024/01/29 by Zhao, Yong-Liang, Gu, Xian-Ming, Oosterlee, Cornelis W.
#65F10 #65L05 #65N22 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2401.16113
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∈ℂ: 1/(1 + α) < |z| < 1/(1 - α)~\rm and~\Re\rm e(z) > 0\, where 0 < α< 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.