R-RESPA method

In the r-RESPA method this operator is decomposed into two or more parts, for example... 

The r-RESPA method has been applied to a variety of systems, including simple model systems [Tuckerman et al. 1992] but also organic molecules [Watanabe and Karplus 1993], fullerene crystals [Procacci and Berne 1994] and also proteins [Humphreys et al. 1994, 1996]. In these studies the reduction in computational time compared with the standard velocity Verlet method varied between 4-5 and 20-40, depending upon the size of the system, without any noticeable loss in accuracy. Other developments of the r-RESPA algorithm include its coupling to the fast multipole method (see Section 6.8.3) [Zhou and Berne 1995]. 

Fig. 1. CPU times (in hours) for 1 ps MD runs for various proteins using three different methods, direct velocity Verlet with a time-step 0.5 fs, r-RESPA with direct evaluation of electrostatic forces and an overall time-step of 4.0 fs, and r-RESPA/TFMM with an overall time-step 4.0 fs (combination of (2,2,2,2) in force breakup).The energy conservation parameter log AE for the three methods are comparable. The CPU time (hours) is for RISC6000 /MODEL 590 computer.

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... 

The problem can further be reduced to two aspects the frequency and computational cost of force calculations. Force calculations, the centres of every MD timestep, range from being very costly to prohibitively costly. The approaches that are currently being pursued in order to solve this problem are quite numerous and include techniques such as the use of multiple timestep methods(r-RESPA) [4], resonance free multiple timestep methods (iso-NHC-RESPA) [5,6], using coarse grained models [3], and other novel approaches. 

Fig. 4. (a) The radial distribution functions goH r), goo r), and gHH r)) of flexible TIP3P liquid water calculated using the NHC-RESPA and ISO-NHC-RESPA methods, (b) The error in the distribution functions as a function of time, where the exact distributions are those that have been generated from a long run with a small time step. The legends report the large time step... 

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]. 

Lecture Notes in Computational Science and Engineering (LNCSE), 24, 146 (2002). Overcoming Instabihties in Verlet-I/R-Respa with the Mollified Impulse Method. 

Recently the particle mesh Ewald method (PME), and a smooth variant of it (SPME), developed by Darden et al., have been described in the literature. " These algorithms are based on Hockney and Eastwood s idea of assigning charges to a mesh according to their real space positions the CPU time savings come from applying the Fast Fourier Transform (FFT) to the particle mesh to accelerate the reciprocal-space calculations of the Ewald sum. The algorithms are found to be of order 0(A log N). This method has been combined with r-RESPA by Procacci, Darden, and Marchi." ... 

Integrators based on r-RESPA, when combined with enhanced methods for calculating long-range electrostatic forces, such as the EMM or SPME schemes, have led to a considerable speedup in the CPU time for large-scale simulations of biomacromolecular solutions. Since r-RESPA is reversible and symplectic such integrators are very stable. Moreover, since r-RESPA is time reversible it can be used... 

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