Stability of asynchronous variational integrators

William Fong

doi:10.1016/j.jcp.2008.05.017

2008-09-01

PRE

William Fong

Eric Darve *

Adrián J. Lew

DOI：10.1016/j.jcp.2008.05.017

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摘要

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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,

Keyword:

symplectic

multiple time step

variational integrator

molecular dynamics