Impulse MTS methods

In LN, the bonded interactions are treated by the approximate linearization, and the local nonbonded interactions, as well as the nonlocal interactions, are treated by constant extrapolation over longer intervals Atm and At, respectively). We define the integers fci,fc2 > 1 by their relation to the different timesteps as Atm — At and At = 2 Atm- This extrapolation as used in LN contrasts the modern impulse MTS methods which only add the contribution of the slow forces at the time of their evaluation. The impulse treatment makes the methods symplectic, but limits the outermost timestep due to resonance (see figures comparing LN to impulse-MTS behavior as the outer timestep is increased in [88]). In fact, the early versions of MTS methods for MD relied on extrapolation and were abandoned because of a notable energy drift. This drift is avoided by the phenomenological, stochastic terms in LN. 

This article is organized as follows Sect. 2 explains why it seems important to use symplectic integrators. Sect. 3 describes the Verlet-I/r-RESPA impulse MTS method, Sect. 4 presents the 5 femtosecond time step barrier. Sect. 5 introduce a possible solution termed the mollified impulse method (MOLLY), and Sect. 6 gives the results of preliminary numerical tests with MOLLY. 

The idea is illustrated by Fig. 1. These equations constitute a readily understandable and concise representation of the widely used Verlet-I/r-RESPA impulse MTS method. The method was described first in [8, 9] but tested... 

An instability of the impulse MTS method for At slightly less than half the period of a normal mode is confirmed by an analytical study of a linear model problem [7]. For another analysis, see [2]. A special case of this model problem, which gives a more transparent description of the phenomenon, is as follows Consider a two-degree-of-freedom system with Hamiltonian p + 5P2 + + 4( 2 This models a system of two springs con-... 

One obvious remedy for this problem is to choose time-step lengths so as to avoid small integer multiples of half-periods of any oscillatory motion. However, it has been demonstrated that the molecular dynamics potential gives rise to motion with a continuum of periods greater than or equal to 10 fs. Furthermore, the energy instability of impulse MTS methods becomes exponentially worse at larger multiples of the half-periods. This rules out the possibility that a fortuitously chosen assortment of impulse multiple time steps longer than 5 fs could yield stable trajectories. 

A number of methods have been proposed to overcome the MTS barrier, including averaging methods that mollify the impulse, allowing time steps of up to 6fs while maintaining the favorably small energy drift attained by impulse MTS methods with 4fs time steps. We will omit here the details of these time-stepping algorithms but point to a reference that explicitly provides implementation details. 

Figure 4 Sensitivity of the impulse MTS method to slow force update intervals. The energy error is essentially unchanged from that of velocity Verlet to an update interval up to 4fs. For larger update intervals, the energy error becomes erratic, with a notable jump at the period of the fastest molecular motion.

Figure 6 Average bond energy as a function of slow force update interval for a Langevin-stabilized extrapolation MTS method. These average energies, taken from long simulations, do not exhibit the sensitivity to the slow force update interval seen with impulse MTS methods in Figure 4.

In this section, we present a simple model problem to illustrate impulse and extrapolation MTS methods for simulations in the constant-energy and constant-temperature regimes. The models are implemented in MATLAB,... 

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