Step methods, multiple time

A straightforward derivation (not reproduced here) shows that the effect of the diree successive steps embodied in equation (b3.3.7), with the above choice of operators, is precisely the velocity Verlet algorithm. This approach is particularly usefiil for generating multiple time-step methods. 

MD, one needs to use multiple time step methods to ensure proper handling of the sprmg vibrations, and there is a possible physical bottleneck in the transfer of energy between the spring system and the other degrees of freedom which must be handled properly [199]. In MC, one needs to use special methods to sample configuration space efficiently [200, 201]. 

In general, multiple-time-step methods increase computational efficiency in a way complementary to multipole methods The latter make use of regularities in space, whereas multiple-time-stepping exploits regularities in time. Figure 2 illustrates the general idea ... 

W. B. Streett, D. J. Tildesley, and G. Saville. Multiple time step methods in molecular dynamics. Mol. Phys., 35 639-648, 1978. 

M. Watanabe and M. Karplus. Dynamics of molecules with internal degrees of freedom by multiple time-step methods. J. Chem. Phys., 99 8063-8074, 1993. 

J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comp. Phys., 109 318-328, 1993. 

Another way to overcome the step-size restriction fc < is to use multiple-time-stepping methods [4] or implicit methods [17, 18, 12, 3). In this paper, we examine the latter possibility. But for large molecular systems, fully implieit methods are very expensive. For that reason, we foeus on the general class of scmi-implicit methods depicted in Fig. 1 [12]. In this scheme. Step 3 of the nth time step ean be combined with Step 1 of the (n - - l)st time step. This then is a staggered two-step splitting method. We refer to [12] for further justification. 

Fig. 1. Schematic for the impulse multiple time stepping method.

Watanabe, M., Karplus, M. Dynamics of Molecules with Internal Degrees of Freedom by Multiple Time-Step Methods. J. Chem. Phys. 99 (1995) 8063-8074 Figueirido, F., Levy, R. M., Zhou, R., Berne, B. J. Large Scale Simulation of Macromolecules in Solution Combining the Periodic Fast Multiple Method with Multiple Time Step Integrators. J. Chem. Phys. 106 (1997) 9835-9849 Derreumaux, P., Zhang, G., Schlick, T, Brooks, B.R. A Truncated Newton Minimizer Adapted for CHARMM and Biomolecular Applications. J. Comp. Chem. 15 (1994) 532-555... 

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

Here we suggest a different approach that propagates the system using multiple step-sizes, i.e., few steps with step-size At are taken in the slow classical part whereas many smaller steps with step-size 5t are taken in the highly oscillatory quantum subsystem (see, for example, [19, 4] for symplectic multiple-time-stepping methods in the context of classical molecular dynamics). Therefore, we consider a splitting of the Hamiltonian H = Hi +H2 in the following way ... 

We have derived time-reversible, symplectic, and second-order multiple-time-stepping methods for the finite-dimensional QCMD model. Theoretical results for general symplectic methods imply that the methods conserve energy over exponentially long periods of time up to small fluctuations. Furthermore, in the limit m —> 0, the adiabatic invariants corresponding to the underlying Born-Oppenheimer approximation will be preserved as well. Finally, the phase shift observed for symmetric methods with a single update of the classical momenta p per macro-time-step At should be avoided by... 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

Constraint and multiple time step methods—Section VIII, this chapter. 

Hamiltonian systems. Thus, one has to treat this non-volume-preserving piece of the integrator a bit more carefully. To ensure numerical stability, higher order reversible integration schemes in conjunction with multiple time step methods are preferred. The details of implementing this scheme are provided in Ref. 28. 

The major limitation of the approaches to multiscale modeling discussed thus far is the timescale. In each of these examples, there are atomic vibrations (on the order of 10 seconds) that need to be followed. This pins down the total simulation time to 0(10 seconds for reasonable calculations. There are many clever multiple time step methods for improving efficiency (e.g., Nakano 1999) by using a quatemion/normal mode representation for atoms that are simply vibrating or rotating, but this buys only a factor of 0(10). 

S. Chin (2004) Dynamical multiple-time stepping methods for overcoming resonance instabilities. J. Chem. Phys. 120, p. 8... 

R. H. Zhou, E. Harder, H. F. Xu, and B. J. Berne, Efficient multiple time step method for use with Ewald and particle mesh Ewald for large biomolecular systems. J. Chem. Phys. 115,2348 2358 (2001). 

Street W B, D Tildesley and G Saville 1978 Multiple Time-step Methods in Molecular Dynarmcs Molecular Physics 35 639-648. 

Watanabe M and M Karplus 1993 Dynamics of Molecules with Internal Degrees of Freedom by Multiple Time-step Methods. Journal of Chemical Physics 99-8063-8074 Widmalm G and R W Pastor 1992. Comparison of Langevin and Molecular Dynamics Simulations. Journal of the Chemical Society Faraday Transactions 88 1747-1754... 

Current developments in simulation techniques include multiple time step methods and more efficient polarizable models. Hardware developments that increase the power of affordable computers will continue to expand the size of the systems treated and the extent of exploring the configurations and dynamics, making the simulations more realistic. 

Biesiadecki JJ, Skeel RD (1993) Dangers of multiple time-step methods. J Comp Phys 109 318-328 Bom M, Huang K (1954) Dynamical Theory of Crystal Lattices. Oxford University Press Bourova E, Parker SC, Richet P (2000) Atomistic simulation of cristobalite at high temperature. Phys Rev B 62 12052-12061... 

Swindoll RD, Haile JM (1984) A multiple time step method for molecular dynamics simulations of fluids of chain molecules. J Comp Phys 53 298-298... 

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