A Parallel-in-Time Implementation of the Numerov Method For Wave Equations

The Numerov method is a well-known 4th-order two-step numerical method for wave equations. It has optimal convergence order among the family of Stormer-Cowell methods and plays a key role in numerical wave propagation. In this paper, we aim to implement this method in a parallel-in-time (PinT) fashion via a diagonalization-based preconditioning technique. The idea lies in forming the difference equations at the N-t time points into an all-at-once system K u = b and then solving it via a fixed point iteration preconditioned by a block alpha-circulant matrix P-alpha, where alpha is an element of (0, 1/2) is a parameter. For any input vector r, we can compute P-alpha(-1) r in a PinT fashion by a diagonalization procedure. To match the accuracy of the Numerov method, we use a 4th-order compact finite difference for spatial discretization. In this case, we show that the spectral radius of the preconditioned iteration matrix can be bounded by alpha/1-alpha provided that the spatial mesh size h and the time step size tau satisfy certain restriction. Interestingly, this restriction on h and tau coincides with the stability condition of the Numerov method. Furthermore, the convergence rate of the preconditioned fixed point iteration is mesh independent and depends only on alpha. We also find that even though the Numerov method itself is unstable, the preconditioned iteration of the corresponding all-at-once system still has a chance to converge, however, very slowly. We provide numerical results for both linear and nonlinear wave equations to illustrate our theoretical findings.