R. D. Skeel and J. J. Biesiadecki. Symplectic integration with variable stepsize. Annals of Numer. Math., 1:191-198, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....work has concentrated on finding methods to handle singularities or widely varying forces without having to evaluate the full force field at every integration step. By basing these methods on potential specific splittings, they can sometimes achieve the e#ect of a variable stepsize method (see [20]) This article takes a di#erent approach in rather trying to adapt the time stepsize directly according to some prescribed rescaling of time. For (1.3) 1.4) using R = #f# corresponds to reparameterization based on the norm of the gradient of the Hamiltonian, hence according to the density ....

R. D. SKEEL AND J. J. BIESIADECKI, Symplectic integration with variable stepsize, Ann. Numer. Math., 1 (1994), pp. 191--198.

....the reach of simulation, even on the fastest computers available. We distinguish two classes of multiscale phenomena. In the rst class, the forces (or potential energies) separate additively into a hierarchy from weak to strong. Sometimes such a hierarchy can be arti cially imposed on the system[9]. This additive decomposition can be used in the design of new methods, as in r RESPA [10] hierarchical variable timestepping [9] the molli ed impulse method [2] and the methods of [8] In the second class of systems, the variables themselves are associated to di erent vibrational scales. Some ....

....class, the forces (or potential energies) separate additively into a hierarchy from weak to strong. Sometimes such a hierarchy can be arti cially imposed on the system[9] This additive decomposition can be used in the design of new methods, as in r RESPA [10] hierarchical variable timestepping [9], the molli ed impulse method [2] and the methods of [8] In the second class of systems, the variables themselves are associated to di erent vibrational scales. Some models could fall into either category, but others do not, since it may not always be easy or natural to partition the variables as ....

Skeel, R.D., and Biesidecki, J.J., Symplectic integration with variable stepsize, Annals of Numerical Mathematics 1, 191-198 (1994).

....variable stepsizes for N body problems. More recently several articles have indicated how regularization methods can be incorporated in a reversible or symplectic framework [20, 15, 13] Another approach to variable stepsize integration, based on a hierarchical splitting, was developed in [26], and we exploit this idea in a somewhat di erent way below, to de ne a smooth switching of the close approaching bodies. We then separate the weaker interactions by splitting, and identify potential close approaches using Verlet lists [30] partitioning the bodies into small groups. The multiple ....

Skeel, R.D. and Biesidecki, J., Symplectic integration with variable stepsize, Annals of Numerical Mathematics, 1, 191-198, 1994.

....variable stepsizes for # body problems. More recently several articles have indicated how regularization methods can be incorporated in a reversible or symplectic framework [20, 15, 13] Another approach to variable stepsize integration, based on a hierarchical splitting, was developed in [26], and we exploit this idea in a somewhat di erent way below, to de ne a ###### ######### of the close approaching bodies. We then separate the weaker interactions by splitting, and identify potential close approaches using ###### ##### [30] partitioning the bodies into small groups. The multiple ....

Skeel, R.D. and Biesidecki, J., Symplectic integration with variable stepsize, ###### ## ######### ###########, #, 191-198, 1994.

.... More recently several articles have indicated how regularization methods can be incorporated in a reversible or symplectic framework [19, 13, 11] DESIGN OF A MECHANICAL N BODY INTEGRATOR 3 Another approach to variable stepsize integration, based on a hierarchical splitting, was developed in [24], and we exploit this idea in a somewhat different way below, to define a smooth switching of the close approaching bodies. We then separate the weaker interactions by splitting, and identify potential close approaches using Verlet lists [28] partitioning the bodies into small groups. The ....

Skeel, R.D. and Biesidecki, J., Symplectic integration with variable stepsize, Annals of Numerical Mathematics, 1, 191, 1994.

....this trajectory using 2 Theta 10 3 timesteps per orbit, with a drift in the above quantities of only one part in 10 7 , over 10 3 revolutions. This seems very impressive. They do not mention any theoretical results on the symplecticness of their method, however. Skeel and Biesiadecki [72] introduce another method for using different timesteps while preserving the symplectic nature of the integrator. They show that, if it is possible to split the Hamiltonian into independent additive terms, then each term may be integrated with a different, but constant, timestep, and the ....

Robert D. Skeel and Jeffrey J. Biesiadecki. Symplectic integration with variable stepsize. Annals of Numerical Mathematics, 1:1--9, 1994.

....to develop variable time step symplectic integrators so that they are competitive : with standard methods : while retaining the good long time behaviour of constant time step symplectic methods . There are several attempts to overcome the difficulties mentioned above. Skeel Biesiadecki [14] decompose the vector field and integrate over a basic time step h different parts with different step sizes (multiple time step method) so that the overall method defines a symplectic transformation. Since the basic step size is not altered, its long time behaviour is like that of a constant ....

R.D. Skeel and J.J. Biesiadecki, Symplectic integration with variable stepsize, Annals of Numerical Mathematics 1 (1994) 191--198.

....work has concentrated on finding methods to handle singularities or widely varying forces without having to evaluate the full force field at every integration step. By basing these methods on potential specific splittings, they can sometimes achieve the effect of a variable stepsize method (see [20]) This article takes a different approach in rather trying to adapt the time stepsize directly according to some prescribed rescaling of time. For (1.3) 1.4) using R = kfk corresponds to reparameterization based on the norm of the gradient of the Hamiltonian, hence according to the density of ....

Skeel, R.D. and Biesiadecki, J.J., Symplectic integration with variable stepsize, Annals of Numer. Math., 1, 191-198, 1994.

....often the mapping to symplectic variables is explicit but its inverse is not; the inverse is required to specify the Hamiltonian in the new variables. This presents algorithmic difficulties that require further investigation. To date the only successful approach is that of Skeel and Biesiadecki [128] and MacEvoy and Scovel [75] They partition the Hamiltonian so that it is a sum of fast and slow terms, the fast terms being zero outside some region of phase space. The slow terms are integrated with a large time step unless that step puts that orbit in the fast region, in which case it must be ....

Skeel, R. D., and J. J. Biesiadecki, Symplectic integration with variable stepsize, Ann. Numer. Math. 1 (1994), 191--198.

....that these forces be treated implicitly. One way to do this is by a splitting of the nonbonded interaction into a slow long range part and a fast short range part. The short range part is chosen to vanish beyond a certain cutoff and it is treated implicitly. For an example of such a splitting see [5]. 3 Linearly implicit methods To solve the equations of an implicit method requires several iterations, which can be costly even if only some forces F 1 are handled implicitly. If we do only one Newton iteration for forces F 1 , the method would be much cheaper but we would sacrifice ....

R. D. Skeel and J. J. Biesiadecki. Symplectic integration with variable stepsize. Annals of Numer. Math., 1:191--198, 1994.

....integrators. McLachlan and Scovel [9] presented the development of efficient symplectic variable stepsize methods as an open problem. Our approach is to apply multiple time stepping techniques to leapfrog [3,14] in a manner that effects variable stepsize, extending results first appearing in [12,16]. The potential energy of the system is partitioned into parts that can be integrated using different fractions of a largest fundamental stepsize. We show how this integration stepsize can be changed while maintaining symplecticness. The method retains the efficiency and important properties of ....

....fundamental stepsize. We show how this integration stepsize can be changed while maintaining symplecticness. The method retains the efficiency and important properties of leapfrog, so is directly applicable to molecular dynamics. The straightforward implementation of the algorithm as described in [12] fails to achieve the efficiencies expected of a variable stepsize algorithm. Here we present two devices for achieving the expected efficiencies. 1) Tables are utilized to evaluate a weighted sum of partial forces for a single interaction at a cost comparable to the direct evaluation of that ....

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R. D. Skeel and J. J. Biesiadecki, Symplectic integration with variable stepsize, Annals of Numer. Math., 191--198, 1994.

....approximation of the full dynamics. Many of the interactions that constitute the collection of forces in a given physical problem can be permanently classified as fast or slow. For interactions of variable speed, it may be computationally e#cient to split them artificially into fast and slow parts [14]. The idea behind the division into W q (q) F (q) is to sample terms of F (q) infrequently and incorporate them into a reduced problem involving W (q) We define a long time step method to be one that samples the slow force at time increments greater than half the period 1 of the fastest ....

R. D. Skeel and J. J. Biesiadecki, Symplectic integration with variable stepsize, Ann. Numer. Math., 1 (1994), pp. 191--198.

....to permit a large timestep whenever r exceeds the cutoff. For example with U(r) 1=r we suggest U soft (r) 3 2 r Gamma1 cut Gamma 1 2 r Gamma3 cut r 2 ; r r cut ; r Gamma1 ; r r cut ; and U hard = U Gamma U soft . Preliminary experiments with this method appear in [40]. 6 Constrained dynamics A traditional approach to taking larger timesteps has been to freeze the bonded motions, thus reducing the stability restriction. This leads to constrained equations of motion which can be discretized, e.g. by the well known SHAKE RATTLE method. In this section we wish ....

Skeel, R. D. and Biesiadecki,J.J., Symplectic integration with variable stepsize, Annals of Numer. Math. 1, 191--198, 1994.

....to permit a large timestep whenever r exceeds the cutoff. For example with U(r) 1=r we suggest U soft (r) 3 2 r Gamma1 cut Gamma 1 2 r Gamma3 cut r 2 ; r r cut ; r Gamma1 ; r r cut ; and U hard = U Gamma U soft . Preliminary experiments with this method appear in [40]. 6 Constrained dynamics A traditional approach to taking larger timesteps has been to freeze the bonded motions, thus reducing the stability restriction. This leads to constrained equations of motion which can be discretized, e.g. by the well known SHAKE RATTLE method. In this section we wish ....

Skeel, R. D. and Biesiadecki,J.J., Symplectic integration with variable stepsize, Annals of Numer. Math. 1, 191--198, 1994.

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R. D. Skeel and J. J. Biesiadecki. Symplectic integration with variable stepsize. Annals of Numer. Math., 1:191-198, 1994.

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R. D. Skeel and J. J. Biesiadecki. Symplectic integration with variable stepsize. Annals of Numer. Math., 1:191--198, 1994.

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R. D. Skeel and J. J. Biesiadecki. Symplectic integration with variable stepsize. Annals of Numer. Math., 1:191--198, 1994. 103

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Robert D. Skeel and Jeffrey J. Biesiadecki. Symplectic integration with variable stepsize. Ann. Num. Math., 1:191--198, 1994.

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Robert D. Skeel and Jeffrey J. Biesiadecki. Symplectic integration with variable stepsize. Ann. Num. Math., 1:191--198, 1994.

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