Multiple time step integrators

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... 

Symplectic Multiple-Time-Stepping Integrators for Quantum-Classical Molecular Dynamics... 

P. F. Batcho, D. A. Case, and T. Schlick, Optimized particle mesh Ewald/multiple time step integration for molecular dynamics simulations. J. Chem. Phys. 115, 4003 4018 (2001). 

Qian, X.L., Schlick, T. Efficient multiple-time-step integrators with distance-based force spHtting for particle-mesh-Ewald molecular dynamics simulations, J. Chem. Phys. 2002,116,5971-83. 

A small number of water models have flexible geometry, such as the TJE [69] and BJH [70] models. The development of multiple time-step integration methods, such as r-RESPA [71] are resulting in greater popularity of models with flexible geometry since flexibility should improve the model s ability to describe more realistically the equilibrium and the dynamical properties of water [72-74]. 

Table 6.2 FORTRAN77 pseudo code of a bitwise time-reversible multiple time-step integrator using Nrespa short (s) time-steps for every long (l) time-step of length Tst. The function But rounds a float to the nearest integer with a precision of the FORTRAN77 function nint rounds off to the nearest integer. The mass of the particles equals 1/Rm.

Since many systems of interest in chemistry have intrinsic multiple time scales it is important to use integrators that deal efficiently with the multiple time scale problem. Since our multiple time step algorithm, the so-called reversible Reference System Propagator Algorithm (r-RESPA) [17, 24, 18, 26] is time reversible and symplectic, they are very useful in combination with HMC for constant temperature simulations of large protein systems. 

Long term simulations require structurally stable integrators. Symplec-tic and symmetric methods nearly perfectly reproduce structural properties of the QCMD equations, as, for example, the conservation of the total energy. We introduced an explicit symplectic method for the QCMD model — the Pickaback scheme— and a symmetric method based on multiple time stepping. 

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. 

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

According to the namre of the empirical potential energy function, described in Chapter 2, different motions can take place on different time scales, e.g., bond stretching and bond angle bending vs. dihedral angle librations and non-bond interactions. Multiple time step (MTS) methods [38-40,42] allow one to use different integration time steps in the same simulation so as to treat the time development of the slow and fast movements most effectively. 

With the propagator written in this way, the equation of motion can be integrated by a multiple time step algorithm in Cartesian coordinates because At and At are different integration time steps (At > At when n> 1). As an example, the force terms are separated into two components... 

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 careful reader should have realized that we choose not to break up this operator with another Trotter factorization, as was done for the extended system case. In practice, one does not multiple-time-step the modified velocity Verlet algorithm because it will, in general, have a unit Jacobian. Thus, one would like the best representation of the operator that can be obtained in closed form. However, even in the case of a modified velocity Verlet operator that has a nonunit Jacobian, multiple-time-stepping this procedure can be costly because of the multiple force evaluations. Generally, if the integrator is stable without multiple-time-step procedures, avoid them. The solution to this first-order inhomogeneous differential equation is standard and can be found in texts on differential equations (see, e.g.. Ref. 53). 

The heart of the rRESPA algorithm is that the equations of motion are integrated by using two different time steps, it is therefore a Multiple Time Step (MTS) method the slow modes (slow forces, 2X3) are integrated with a larger time step. At, whereas the fast modes (fast forces and velocities, 2X1, 2X2) with a smaller time step, St(St = Atjri). In this case the evolution operator becomes [16]... 

The general plan for a multiple time-step numerical method is that will be evaluated at every step of the integration at time increments h, while F will be evaluated less frequently, typically at time increments xh where X > 1 is an integer. The key question is How should be incorporated into the numerical dynamics In the original work of Streett et al., " the slow force on particle i was approximated by a truncated Taylor series at each step 0 < < x, between updates at steps t and t + xh ... 

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