Multiple timestepping

The stringency of the timestep threshold and the failure of implicit methods to address the issue have led to alternative proposals for ameliorating the instability in molecular dynamics, in particular the method of multiple timestepping which we next describe. Variants of this method were described in a series of papers in the early 1990s by Tuckerman and Berne [374] and Tuckerman et al. [375, 377]. A simultaneous development by Grubmiiller et al. [158] most closely matches the presentation given here. The first resonance analysis is due to Biesiadecki and Skeel [35]. 

In many molecular dynamics applications, the system may be decomposed as H q.p) = p M p/2 + Usiq) + where Us, is a soft potential, in which the natural motion would be relatively slow, whereas f/p is a much stronger term which would give rise to relatively rapid motion. We further suppose Us (with its gradient) is expensive to compute, perhaps because it involves many terms, whereas 17f is relatively simple. Because U requires small timesteps, we think of separating the calculation of a timestep, typically in a symmetric form. Let denote the map associated to the Verlet method for the system witliout Us, using a stepsize fr. 

One step of the multiple timestepping algorithm is then implemented as follows Reversible RESPA Algorithm 

Multiple timestepping is widely used for efficiency purposes in systems with expensive force computations, but it should be used cautiously. To illustrate the potential dangers of multiple timestepping, one need only consider its application to a linear one-DOF model problem 

The frequency is therefore S2 = Vl + In the limit of small inner step, we solve 

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An elegant derivation of the Verlet-type algorithms has been given by Tuckerman et al. [36] and is useful in multiple timestep implementations. 

The heightened appreciation of resonance problems, in particular, has been quite recent [63, 62], and contrasts the more systematic error associated with numerical stability that grows systematically with the discretization size. Ironically, resonance artifacts are worse in the modern impulse multiple-timestep methods, formulated to be symplectic and reversible the earlier extrapolative variants were abandoned due to energy drifts. 

E. Barth and T. Schlick. Extrapolation versus impulse in multiple-timestepping schemes II. linear analysis and applications to Newtonian and Langevin dynamics. J. Chem. Phys., 109 1632-1642, 1998. 

T. Bishop, R. D. Skeel, and K. Schulten. Difficulties with multiple timestepping and the fast multipole algorithm in molecular dynamics. J. Comput. Chem., 18 1785-1791, 1997. 

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. 4.3 Eigenvalues of the simple multiple timestepping method. The eigenvalues coil around the surface of the cylinder A = 1 except at exceptional points near kn Q, where k h. Near these points of resonance, exponential instabilities are present...

We can easily solve for the two eigenvalues as functions of h and graph them as points in three dimensions x = Re(A), y = Im(A), z = h (Fig. 4.3). Such resonances will also occur at integer multiples of n/Q, and we can therefore expect many small intervals in the stepsize where the multiple timestepping method is unstable. When a large system involves a combination of many different interactions, the intervals of instability would expect to proliferate from all the different natural frequencies. 

In each of the mollified multiple timestepping methods, since only the potential energy is modified, and by a smooth mapping of positions, the forces derived by differentiating U are still conservative, and, just as with the MTS method, the scheme remains symplectic. 

One approach to addressing these stiff terms is to use a multiple timestepping method. A more direct approach is to introduce constraints to simply remove the stiff bond stretches. This makes sense if (a) the motion of the constrained system can be simulated using larger timesteps and (b) the rigidification of certain vibrational terms does not significantly alter the thermodynamic or dynamic properties of interest to the modeller. 

Fig. 7.13 We consider the stability of a multiple timestepping scheme incorporating Langevin dynamics, with potential energy function (7.47) choosing 12 = 3. Each pixel represents a simulation undertaken with parameters (t, y), for t = Qh. The color of a pixel indicating the absolute error in (q )f where white pixels indicate instability of the method. The plotted line gives the boundary to the region in parameter space where (7.49) is satisfied...

It is important to emphasize the tremendous complexity of molecular dynamics in relation to the simple model scenario presented here. In [22, 29], multiple timestepping was used effectively to treat large scale systems where the force spUtting is defined in the context of particle-mesh Ewald long-ranged force computations. We also point out that where additional constraints are present, the prospects for multiple timestepping may be significantly better, as, in the previously mentioned work [118] and in Sect. 8.6 of the next chapter, where isokinetic constraints are used to stabilize the system. 

Multiple Timestepping Using the Stochastic Isokinetic Formulation... 

As explained in Chap. 4, the resonances in multiple timestepping restrict the size of the large (outer) stepsize usable in simulation. In the case of fuUy flexible water the maximum outer stepsize that can be used for stable RESPA simulation is typically found to be less than 5 fs [245, 329]. A multiple timestepping method for the SPME method can be based on the decomposition... 

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