Stepping methods

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]. 

Bofill J M 1994 Updated Hessian matrix and the restricted step method for locating transition structures J. Comput. Chem. 15 1... 

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. 

B. Garcia-Archilla, J.M. Sanz-Serna, and R.D. Skeel. Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comp., 1996. To appear, [Also Tech. Kept. 1996/7, Dep. Math. Applic. Comput., Univ. Valladolid, Valladolid, Spain). 

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... 

Garcia-Archilla, B., Sanz-Serna, J.M., Skeel, R.D. Long-Time-Steps Methods for Oscillatory Differential Equations. SIAM J. Sci. Comput. (to appear)... 

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... 

A different long-time-step method was previously proposed by Garci a-Archilla, Sanz-Serna, and Skeel [8]. Their mollified impulse method, which is based on the concept of operator splitting and also reduces to the Verlet scheme for A = 0 and admits second-order error estimates independently of the frequencies of A, reads as follows when applied to (1) ... 

HyperChein uses the steepest descent by steps method. New points are found 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] ... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

After application of the 6 time-stepping method (see Chapter 2, Section 2.5) and following the procedure outlined in Chapter 2, Section 2.4, a functional representing the sum of the squares of the approximation error generated by the finite element discretization of Equation (4.118) is formulated as... 

The remaining terms in equation set (4.125) are identical to their counterparts derived for the steady-state case (given as Equations (4.55) to (4.60)). By application of the 9 time-stepping method, described in Chapter 2, Section 2.5, to the set of first-order ordinary differential equations (4.125) the working equations of the solution scheme are obtained. The general form of tliese equations will be identical to Equation (2.111) in Chapter 2,... 

Application of the previously described 6 time-stepping method to Equation (5.11) gives... 

The Leimgruber-Batcho synthesis is a two-step method which provides indoles that arc substituted only in the benzene ring. The method was initially disclosed in a patent[l] and a representative procedure is available in Organic Syntheses[2]. A review of the reaction is available[3]. The reaction involves... 

The steepest descent by steps method may provide a reasonably good method to begin an optimization when the starting point is far from the minimum. However, it converges slowly near the minimum and it is principally recommended only to initiate optimization when the starting point is particularly bad. 

A successful synthesis of novel, soluble aromatic Pis involving 3,4-bis-(4-aminophen5l)-2,5-diphen5lfuran by polymerization with aromatic tetracarboxyhc dianhydrides through the conventional two-step method has been reported (6) (Fig. I). 

A general one-step method for preparation of primary and secondary nitroparaffins from amines by oxidation with y -chloroperbenzoic acid in 1,2-dichloroethane has been reported (68). This method is particularly useful for laboratory quantities of a wide variety of nitroparaffins because a large number of amines are readily available from ketones by oxime reduction and because the reaction is highly specific for nitroparaffins. 

The reaction of lithio derivatives with appropriate electrophiles has been utilized in the preparation of alkyl, aryl, acyl and carboxylic acid derivatives. Representative examples of these conversions are given in Scheme 79. Noteworthy is the two-step method of alkylation involving reaction with trialkylborane followed by treatment with iodine (78JOC4684). 

The phenyl group became a practical protective group for carboxylic acids when Sharpless published a mild, effective one-step method for its conversion to a carboxylic acid. It has recently been used in a synthesis of the amino acid statine, where it served as a masked or carboxylic acid equivalent. ... 

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