IMEX Runge-Kutta Parareal for Non-diffusive Equations

Abstract

Parareal is a widely studied parallel-in-time method that can achieve meaningful speedup on certain problems. However, it is well known that the method typically performs poorly on non-diffusive equations. This paper analyzes linear stability and convergence for IMEX Runge-Kutta Parareal methods on non-diffusive equations. By combining standard linear stability analysis with a simple convergence analysis, we find that certain Parareal configurations can achieve parallel speedup on non-diffusive equations. These stable configurations possess low iteration counts, large block sizes, and a large number of processors. Numerical examples using the nonlinear Schrödinger equation demonstrate the analytical conclusions.

Appendices

Appendix 1: Infinity Norm of the Parareal Iteration Matrix E

Let $\mathbf {A}(\gamma )$ be the lower bidiagonal matrix

$$\begin{aligned} \mathbf {A}(\gamma ) = \left[ \begin{array}{cccc} 1 &{} &{} &{} \\ \gamma &{} 1 &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} \gamma &{} 1 \end{array}. \right] \end{aligned}$$

Lemma 1

The inverse of $\mathbf {A}(\gamma )$ is given by

$$\begin{aligned} \mathbf {A}^{-1}_{i,j}(\gamma ) = {\left\{ \begin{array}{ll} (-\gamma )^{i-j} &{} j \le i \\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

Proof

For convenience, we temporarily drop the $\gamma$ so that $\mathbf {A}=\mathbf {A}(\gamma )$, then

$$\begin{aligned} \left( \mathbf {A} \mathbf {A}^{-1} \right) _{ij} = \sum _{k=1}^{{N_p}+1} \mathbf {A}_{ik}\mathbf {A}_{kj}^{-1}&= {\left\{ \begin{array}{ll} 0 &{} j> i\\ \mathbf {A}_{ii}\mathbf {A}_{ii}^{-1} &{} i=j \\ \mathbf {A}_{ii}\mathbf {A}_{ij}^{-1} + A_{i, i-1} A_{i-1,j}^{-1} &{} j<i \\ \end{array}\right. } \\&= {\left\{ \begin{array}{ll} 0 &{} j > i\\ 1 &{} i=j \\ (-\gamma )^{i-j} + \gamma (-\gamma )^{i-1-j} &{} j<i \end{array}\right. } \\&= {\left\{ \begin{array}{ll} 1 &{} i=j, \\ 0 &{} \text {otherwise} . \end{array}\right. } \end{aligned}$$

Lemma 2

The product of $\mathbf {A}(\omega )\mathbf {A}^{-1}(\gamma )$ is

$$\begin{aligned} \left( \mathbf {A}(\omega ) \mathbf {A}^{-1}(\gamma ) \right) _{ij} = {\left\{ \begin{array}{ll} 0 &{} j > i \\ 1 &{} i = j \\ (-\gamma )^{i-j-1}(\omega - \gamma ) &{} j < i \end{array}\right. } \end{aligned}$$

Proof

$$\begin{aligned} \left( \mathbf {A}(\omega ) \mathbf {A}^{-1}(\gamma ) \right) _{ij}&= \sum _{k=1}^{{N_p}+1} \mathbf {A}_{ik}(\omega )\mathbf {A}_{kj}^{-1}(\gamma ) \\&= {\left\{ \begin{array}{ll} 0 &{} j> i\\ \mathbf {A}_{ii}(\omega ) \mathbf {A}_{ii}^{-1}(\gamma ) &{} i=j \\ \mathbf {A}_{ii}(\omega ) \mathbf {A}_{ij}^{-1} + \mathbf {A}_{i, i-1}(\omega ) \mathbf {A}_{i-1,j}^{-1} (\gamma ) &{} j<i \end{array}\right. } \\&= {\left\{ \begin{array}{ll} 0 &{} j > i\\ 1 &{} i=j \\ (-\gamma )^{i-j} + \omega (-\gamma )^{i-1-j} &{} j<i \end{array}\right. } \end{aligned}$$

Lemma 3

The infinity norm of the matrix $M(\omega , \gamma ) {=} \mathbf {I} {-} \mathbf {A}(\omega )\mathbf {A}^{-1}(\gamma ) {\in } \mathbb {R}^{{N_p}+1, {N_p}+1}$ is

$$\begin{aligned} \Vert \mathbf {M}(\omega , \gamma ) \Vert _{\infty } = \frac{1 - |\gamma |^{{N_p}}}{1 - |\gamma |} |\gamma - \omega |. \end{aligned}$$

Proof

Using Lemma 2, the jth absolute column sum of $M(\omega , \gamma )$ is

$$\begin{aligned} c_j = \sum _{k=j+1}^{{N_p}+1} |(-\gamma )^{k-j-1}(\omega - \gamma )| = \sum _{k=1}^{{N_p}- j} |(-\gamma )^{k}||(\omega - \gamma )| \end{aligned}$$

It follows that $\max _j c_j = c_1$, which can be rewritten as

$$\begin{aligned} \frac{1 - |\gamma |^{{N_p}}}{1 - |\gamma |} |\gamma - \omega |. \end{aligned}$$

Appendix 2: Additional Stability and Convergence Overlay Plots

Figures 8, 9, and 10 show stability and convergence overlay plots for Parareal. The following three figures show stability and convergence overlay plots for Parareal configurations with: ${N_T}= 2048$, IMEX-RK4 as the fine integrator, and three different coarse integrators. These additional figures supplement Fig. 3 and show the effects of changing the course integrator.

Fig. 8

Stability and convergence overlay plots for Parareal configurations with a block size of ${N_T}=2048$ and IMEX-RK1, IMEX-RK4 as the coarse and fine integrators

Fig. 9

Stability and convergence overlay plots for Parareal configurations with a block size of ${N_T}=2048$ and IMEX-RK2, IMEX-RK4 as the coarse and fine integrators

Fig. 10

Stability and convergence overlay plots for Parareal configurations with a block size of ${N_T}=2048$ and IMEX-RK4, IMEX-RK4 as the coarse and fine integrators