薛定谔方程的时间并行模拟
Time-parallel simulation of the Schrödinger Equation
摘要：
量子系统含时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该解决方案是具有挑战性的，并且要求使用现代高性能计算系统。虽然经典的空间并行化技术可以减少每个时间步长的运行时间，但新颖的时间并行积分器暴露了时间域中的并行性。然而，它们对于像薛定谔方程这样的波动型问题并不十分适用。一个值得注意的例外是指数积分器的有理逼近。在本文中，我们推导出一个有效的变体，这种方法适用于复值薛定谔方程。使用Faber-Carathéodory-Fejér近似，这种变体已经是一种快速串行，特别是一种有效的时间并行积分器。它可以用来增强经典的空间并行化，我们沿着两个具有挑战性的，现实的例子，我们的方法的效率和有效性。
英文摘要：
The numerical simulation of the time-dependent Schrödinger 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 Schrödinger 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 Schrödinger equation. Using the Faber-Carathéodory-Fejér 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.
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