Multiple-time-step algorithms

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. 

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... 

M. Tuckermann, B.J. Berne and G.J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys., 97 (1992) 1990 P. Procacci and B.J. Berne, Computer simulation of solid C60 using multiple time-step algorithms, J. Chem. Phys., 101 (1994) 2421. 

C. J. Mundy, J. I. Siepmann, and M. L. Klein, J. Chetn. Phys., 102, 3376 (1995). Calculation of the Shear Viscosity of Decane Using a Reversible Multiple Time-Step Algorithm. 

The combination of the multiple time step algorithm and PME[27] makes the simulation of large size biomolecular systems such as membrane proteins extremely efficient and affordable even for long time spans. Furthermore, it does not involve any uncontrolled approximation and is entirely consistent with periodic boundary conditions. 

Pearce L L and S C Harvey 1993. Langevin Dynamics Studies of Unsaturated Phospholipids in a Membrane Environment. Biophysical Journal 65 1084-1092 Procacci P and B Berne 1994 Computer Simulation of Solid C o Using Multiple Time-step Algorithms Journal of Chemical Physics 101-2421-2431. 

Leimkuhler, B., Margul, D., Tuckerman, M. Stochastic, resonance-free multiple time-step algorithm for molecular dynamics with very large time steps. Mol. Phys. Ill, 3579-3594 (2013). doi 10.1080/00268976.2013.844369... 

IV. LIOUVILLE FORMULATION OF EQUATIONS OF MOTION—MULTIPLE TIME STEP ALGORITHMS... 

One of the issues in dynamical multiscale coupling is the tailoring of the time step to the different subdomains. If the same time step is used in both the atomistic and the continuum regions, computations will be wasted in the continuum model. However, if in the hand-shake region the size of the FEM elements is reduced to coincide with the individual atoms, it is difficult to tailor the time step. Therefore, the authors of the ODD method chose to use a uniform mesh for the continuum domain, so that a much larger time step could be used in the continuum model than in the atomistic one. A description of such a multiple-time-step algorithm is provided in the paper. 

Molecular dynamics (MD) is the most widely used computational method to study the kinetic and thermodynamic properties of atomic and molecular systems.These properties are obtained by solving the microscopic equations of motion (Eq. [1]) for the system under consideration. The multiple time-step algorithms discussed earlier have extended the time scale that can be reached, but, the gain is still insufficient for the study of many processes for many systems, such as biomolecules, this simulation time is inadequate to study large conformational changes or to study rare but important events as examples. 

Molecular Dynamics Simulations of Solvated Proteins via a Multiple Time Step Algorithm. 

Recently, rigorous multiple time step algorithms have been invented, which can significantly augment the ratio of simulated time to CPU time. 

Appendix B Bitwise time-reversible multiple time-step algorithm... 

---