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

Abstract

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 Störmer-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 ${\mathscr {K}}{{\varvec{u}}}={{\varvec{b}}}$ and then solving it via a fixed point iteration preconditioned by a block $\alpha$-circulant matrix ${\mathscr {P}}_\alpha$, where $\alpha \in (0,\frac{1}{2})$ is a parameter. For any input vector ${{\varvec{r}}}$, we can compute ${\mathscr {P}}_{\alpha }^{-1}{{\varvec{r}}}$ 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 $\frac{\alpha }{1-\alpha }$ 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.