Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schr\"odinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations.

翻译：摘要：Parareal是一种著名的并行时间算法，通过在并行迭代中结合粗网格和细网格传播算子，能够实现大规模并行计算，相较于串行时间步进方法可显著减少计算时间。然而，与许多并行时间方法类似，当应用于非扩散型方程（如双曲系统或色散非线性波动方程）时，该算法可能无法收敛。本文探讨了在Parareal迭代中使用指数积分器的可能性。指数积分器能够在大步长（粗网格传播算子所用步长）下解析快速传播的波动，因此成为Parareal算法的理想候选对象。本文首先介绍了指数型Parareal积分器，随后通过涉及非线性薛定谔方程的数值实验进行激励性验证。基于这些实验，采用线性分析方法近似评估指数型Parareal迭代在非线性问题上的稳定性和收敛特性。最后，本文通过两个附加数值实验（涉及色散型Kadomtsev-Petviashvili方程和双曲型Vlasov-Poisson方程）证明：在求解特定非扩散型方程时，指数型Parareal方法相较于串行指数积分器能显著缩短求解时间。