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Stability of asynchronous variational integrators

机译：异步变分积分器的稳定性

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The adoption of multiple time step integrators can provide substantial computational savings for mechanical systems with multiple time scales. However, the scope of these savings may be limited by the range of allowable time step choices. In this paper we analyze the linear stability of the fully asynchronous methods termed AVI, for asynchronous variational integrators. We perform a detailed analysis for the case of a one-dimensional particle moving under the action of a soft and a stiff quadratic potential, integrated with two time steps in rational ratios. In this case, we provide sufficient conditions for the stability of the method. These generalize to the fully asynchronous AVI case the results obtained for synchronous multiple time stepping schemes, such as r-RESPA, which show resonances when the larger time step is a multiple of the effective half-period of the stiff potential. Additionally, we numerically investigate the appearance of instabilities. Based on the experimental observations, we conjecture the existence of a dense set of unstable time steps when arbitrary rational ratios of time steps are considered. In this way, unstable schemes for arbitrarily small time steps can be obtained. However, the vast majority of these instabilities are extremely weak and do not present an obstacle to the use of these integrators. We then applied these results to analyze the stability of multiple time step integrators in the more complex mechanical systems arising in molecular dynamics and solid dynamics. We explained why strong resonances are ubiquitously found in the former, while rarely encountered in the latter. Finally, in this paper we introduce a formulation of AVI that highlights the symplectic nature of the algorithm, complementing those introduced earlier by other authors. (c) 2008 Elsevier Inc. All rights reserved,
机译：采用多个时间步长积分器可以为具有多个时标的机械系统节省大量计算资源。但是，这些节省的范围可能会受到允许的时间步长选择范围的限制。在本文中，我们针对异步变分积分器分析了称为AVI的完全异步方法的线性稳定性。我们对一维粒子在软和刚性二次势的作用下移动的情况进行了详细分析，并结合了两个时间步长的合理比率。在这种情况下，我们为该方法的稳定性提供了充分的条件。这些将通用异步AVI情况推广到完全同步的多个时间步进方案（如r-RESPA）获得的结果，当较大的时间步长是刚性势能的有效半周期的倍数时，该结果显示出共振。此外，我们在数值上研究了不稳定性的出现。基于实验观察，我们推测当考虑任意合理的时间步长比率时，存在一组密集的不稳定时间步长。以这种方式，可以获得任意小的时间步长的不稳定方案。但是，这些不稳定性中的绝大多数非常弱，并且不构成使用这些积分器的障碍。然后，我们将这些结果应用于分析在分子动力学和固体动力学中产生的更复杂的机械系统中多个时间步积分器的稳定性。我们解释了为什么在前者中普遍存在强烈的共鸣，而在后者中却很少遇到。最后，在本文中，我们介绍了AVI的公式，该公式突出了算法的辛性质，是对其他作者较早前介绍的算法的补充。 （c）2008 Elsevier Inc.保留所有权利，

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