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

Yafei Sun, Shu-Lin Wu, Yingxiang Xu

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 $$N_t$$ N t time points into an all-at-once system $${\mathscr {K}}{{\varvec{u}}}={{\varvec{b}}}$$ K u = b and then solving it via a fixed point iteration preconditioned by a block $$\alpha $$ α -circulant matrix $${\mathscr {P}}_\alpha $$ P α , where $$\alpha \in (0,\frac{1}{2})$$ α ∈ ( 0 , 1 2 ) is a parameter. For any input vector $${{\varvec{r}}}$$ r , we can compute $${\mathscr {P}}_{\alpha }^{-1}{{\varvec{r}}}$$ P α - 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 $$\frac{\alpha }{1-\alpha }$$ α 1 - α provided that the spatial mesh size h and the time step size $$\tau $$ τ satisfy certain restriction. Interestingly, this restriction on h and $$\tau $$ τ 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 $$\alpha $$ α . 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.