The numerical simulation of the time-dependent Schrodinger equation for quantum systems is a very active research topic. Yet, resolving the solution sufficiently in space and time is challenging and mandates the use of modern high-performance computing systems. While classical parallelization techniques in space can reduce the runtime per time-step, novel parallel-in-time integrators expose parallelism in the temporal domain. They work, however, not very well for wave-type problems such as the Schrodinger equation. One notable exception is the rational approximation of exponential integrators. In this paper we derive an efficient variant of this approach suitable for the complex-valued Schrodinger equation. Using the Faber-Caratheodory-Fejer approximation, this variant is already a fast serial and in particular an efficient time-parallel integrator. It can be used to augment classical parallelization in space and we show the efficiency and effectiveness of our method along the lines of two challenging, realistic examples.

中文翻译：

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薛定谔方程的时间平行模拟

量子系统的瞬态薛定谔方程的数值模拟是一个非常活跃的研究课题。然而，在空间和时间上充分解决解决方案具有挑战性，并且要求使用现代高性能计算系统。虽然空间中的经典并行化技术可以减少每个时间步的运行时间，但新颖的时间并行积分器暴露了时间域中的并行性。然而，它们对于诸如薛定谔方程之类的波型问题的效果不是很好。一个值得注意的例外是指数积分器的有理近似。在本文中，我们推导出适用于复值薛定谔方程的这种方法的有效变体。使用 Faber-Caratheodory-Fejer 近似，这个变体已经是一个快速串行，特别是一个高效的时间并行积分器。它可用于增强空间中的经典并行化，我们通过两个具有挑战性的现实示例展示了我们方法的效率和有效性。