A Generalized Block Circulant Preconditioner for Crank-Nicolson All-at-Once Systems with Applications to Option Pricing PDEs
The Crank--Nicolson (CN) method is a widely used time integration scheme for evolutionary partial differential equations (PDEs) arising in various scientific and engineering disciplines. Since the numerical solution at each time level depends on the solution at the previous time level, the resulting discretization is inherently sequential and therefore difficult to parallelize in time. In this paper, we develop an all-at-once formulation of the CN discretization together with a generalized block circulant preconditioner that enables an efficient parallel-in-time solution within a Krylov subspace framework. We establish a detailed spectral analysis of the preconditioned system, proving that most eigenvalues are equal to $1$, while the remaining eigenvalues are confined to the annulus: \begin{equation*} \left\{ z\in\mathbb{C}: \frac{1}{1+α}<|z|<\frac{1}{1-α}, \ \Re(z)>0 \right\}, \end{equation*} 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($m$) method achieves a fast convergence rate independent of discretization stepsizes from the residual point of view. Finally, we verify both theoretical findings and the efficacy of the proposed preconditioner via numerical experiments on financial option pricing PDEs (even with variable coefficients).