J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109(2):318-328, Dec. 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Methods Here we suggest a different approach that propagates the system using multiple step sizes, i.e. few steps with step size Deltat are taken in the slow classical part whereas many smaller steps with stepsize ffit 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 = H 1 H 2 in the following way: and H 2 = H(q) U cl (q) Let us write down the corresponding differential equations. First for H ....

J.J. Biesiadecki and R.D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109:318--328, 1993.

....= p n Gamma Deltat 2 r q U(q n ) p n 1 = p n 1=2 Gamma Deltat 2 r q U(q n 1 ) However, if K i AE 1, a small step size Deltat of order ffl with ffl Gamma2 = max i=1; m K i ; has to be used. This problem can be avoided by either using multiple timestepping methods [8] 15] [4] or by replacing the bond stretching and bond angle bending modes by holonomic constraints [1] which leads to the constrained Hamiltonian system d dt q = M Gamma1 p ; 1) d dt p = Gammar q U(q) Gamma r q g(q) 2) 0 = g(q) 3) which can be discretized by the symplectic SHAKE or RATTLE ....

....and implies additional long range force field evaluations per time step. However, noting that the only significant contributions to the modified constraint function g come from nearest neighborhood interactions, the potential energy U l can be split as in multiple time stepping methods [8] 15] [4] and only the nearest neighborhood interactions are included in the evaluation of g. In fact, the main disadvantage of the formulation lies in the fact that it requires the computation of the gradient of g and thus the computation of the Hessian of U . 2 A Modified Potential Energy Function ....

Biesiadecki, J.J. and Skeel, R.D., Dangers of Multiple-Time-Step Methods, J. Comput. Phys. 109, 318--328, 1993.

.... fast period limits the outer timestep to somewhat less than T min =2 in standard protocols (i.e. half the period of the fastest motion, which is around 10 fs) Though the first applications attributed these disturbances to general inaccuracies, they were later recognized as resonance artifacts [10, 11]. These artifacts have been analyzed in connection with implicit integration schemes such as implicit midpoint [12] and related integrators [13, 14] and with MTS (or force splitting) schemes [10, 11] Impulse MTS schemes [6, 7] are particularly vulnerable to resonances since relatively large ....

.... these disturbances to general inaccuracies, they were later recognized as resonance artifacts [10, 11] These artifacts have been analyzed in connection with implicit integration schemes such as implicit midpoint [12] and related integrators [13, 14] and with MTS (or force splitting) schemes [10, 11]. Impulse MTS schemes [6, 7] are particularly vulnerable to resonances since relatively large energy pulses are introduced to the governing dynamics equations when the slow forces are evaluated. These large pulses in turn lead to incorrect physical behavior of the system, such as overstretching ....

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J.J. Biesiadecki and R.D. Skeel. Dangers of multiple time step methods. J. Comp. Phys., 109:318--328, 1993.

.... Methods Here we suggest a different approach that propagates the system using multiple step sizes, i.e. few steps with step size Deltat are taken in the slow classical part whereas many smaller steps with step size ffit are taken in the highly oscillatory quantum subsystem (see, for example, [4] for symplectic multiple time stepping methods in the context of classical molecular dynamics) Therefore, we consider a splitting of the Hamiltonian H = H 1 H 2 in the following way: H 1 = p T M Gamma1 p 2 and H 2 = H(q) U cl (q) Let us write down the corresponding ....

J.J. Biesiadecki and R.D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109:318--328, 1993.

....(For example, in molecular dynamics simulations without cut off, up to 90 of the interparticle forces are weak forces in the above sense. 5 Numerical Example As a numerical test example we simulated a two dimensional frozen argon crystal. We used the same configuration as described in [2]; i.e. six argon atoms arranged symmetrically around a center atom with Lennards Jones potential between each atom pair. The potential energy of the system is V (q) 1 2 6 X i=1 7 X j=i 1 OE(jjr j Gamma r i jj 2 ) where OE(r 2 ) 8ffl ffi r 12 Gamma ffi r 6 : ....

....= 0:2383 kcal mol, ffi = 3:405 A, and particle mass m = 39:95 amu. The initial positions of the particles are slightly perturbed about those for lowest potential energy. They are given low initial velocities such that the total momentum of the system is zero and the initial temperature is 22:72 K [2]. The system was simulated using our enhanced energy method method of Section 4. The simulation was run for a time of t tot = 1:0 nanosecond. The computed average kinetic energy hE kin i and its standard deviation h( DeltaE kin ) 2 i 1=2 = h(E kin Gamma hE kin i) 2 i 1=2 for step sizes ....

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J.J. Biesiadecki and R.D. Skeel, Danger of multiple-time-step methods, J. Comput. Phys. 109, 318--328, 1993.

....p) p t M Gamma1 p 2 V 1 (q) 1 ffl V 2 (q) where ffl 0 is a small number. This type of Hamiltonian systems arise, for example, in molecular dynamics simulations [1] 27] Any other symplectic, second order method for unconstrained problems, such as the multiple time step methods [8] etc. can now be generalized to the constrained case along the same lines. This is especially useful whenever one wishes to exploit the special structure of the Hamiltonian H. Based on these second order schemes, methods of higher order can be constructed, e.g. by Yoshida s method [31] or, more ....

....common that they are not suited for the integration of Hamiltonians (2) with V (q) V 1 (q) 1 ffl V 2 (q) where ffl 0 is a small parameter. Such problems arise, for example, in the context of molecular dynamics simulations [27] In this case one could either use multiple timestep methods [8] or the implicit midpoint rule as the basic integrator in (24) Another possibility would be to use the following combination of the midpoint and Verlet scheme: Discretize the Hamiltonian H s (q; p) p t M Gamma1 p 2 1 ffl V 2 (q) by the implicit midpoint rule and call the resulting ....

Biesiadecki, J.J. and Skeel, R.D., Danger of Multiple-Time-Step Methods, J. Comput. Phys., 109(1993), 318--328.

....methods) that use different time steps to integrate fast and slow solution components. Such an approach has been developed further by Gear Wells [3] Gunther Rentrop [4] and Skelboe Anderson [11] In a closely related approach, known as multiple time stepping , see Biesiadecki Skeel [1], one does not split solution components, but instead the right hand side function as a sum of fast and slowly changing functions which are then evaluated with different rates. A difficulty with the existing multirate techniques is that they assume a clear cut partition of the system into fast and ....

J.J. Biesiadecki, R.D. Skeel (1993): Dangers of multiple time step methods. J. Comp. Phys., 109, pp. 318-328.

....method. The main disadvantage of this approach arises when the limit constrained system is different from (4) as mentioned in the introduction and demonstrated in x5 for our second model problem. Another way to overcome the step size restriction k is to use multiple timestepping methods [4] or implicit methods [15, 16, 11, 3] In this paper, we examine the latter possibility. But for large molecular systems, fully implicit methods are very expensive. For that reason, we focus on the following general class of semi implicit methods [11] Semi Implicit Integrator Step 1. p 1;n = ....

J.J. Biesiadecki and R.D. Skeel. Danger of multiple-time-step methods. J. Comput. Phys., 109:318--328, 1993.

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J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109(2):318--328, Dec. 1993.

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J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109:318--328, 1993.

....; P n 1 ) and uses the fact that the composition of symplectic maps is also symplectic and the Jacobian matrix of F (R) is symmetric. The Verlet method can be generalized to multiple time steps in various ways [5] most of which destroy the symplecticness. There is a way to retain symplecticness [1], which we describe for multiple stepsizes h and Nh. Suppose we express V = V hard V soft and correspondingly partition the force vector. Then define 4 R.D. Skeel, J.J. Biesiadecki Symplectic integration with variable stepsize F n = F hard n NF soft n ; n a multiple of N; F ....

....4 R.D. Skeel, J.J. Biesiadecki Symplectic integration with variable stepsize F n = F hard n NF soft n ; n a multiple of N; F hard n ; otherwise. This retains the second order accuracy of the Verlet method as well as its symplectic property. It is called the Verlet I method in [5, 1]. In order to effect variable stepsize, we do an artificial partitioning of an interaction V into a short range interaction and a smooth long range interaction. For example, consider the potential energy V (r) C r ; such as might occur due to gravitational or electrostatic attraction. We ....

J.J. Biesiadecki and R.D. Skeel, Dangers of multiple-time-step methods, J. Comput. Phys. to appear.

.... in [5, 7] but these writings express little enthusiasm for the method because of the possibility of resonance if the period h of the impulse should happen to coincide with a natural frequency of the reduced system M(d 2 dt 2 )q = W q (q) The resonance is demonstrated experimentally in [1]. Also, molecular dynamics experiments in [4] seem to indicate that the step size has to be less than the resonance value, which is 9 10 fs for fully flexible classical mechanics models of molecules. Other experiments [6, 8] show the inferiority of the impulse method (Verlet I) in a Langevin ....

....# 8#. This shows a degraded accuracy when LONG TIME STEPS FOR OSCILLATORY EQUATIONS 943 Fig. 10. Error in energy for LongAverage, LinearAverage, ShortAverage, and Impulse methods; testing for instability of type 3. 0 5 10 15 20 25 30 35 10 3 10 2 10 1 Omega1 2 springs Filter One: p0=[1 1 1 1] 2 sqrt(2) h=0.5 . h=0.25 . h=0.125 Fig. 11. Maximum error in position versus# 1 for Impulse method for each of three di#erent step sizes. h# 1 is near 2# and, furthermore, that the maximum position error (over all possible choices of# 1 ) exhibits an O(h) behavior. This is an order ....

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J. J. Biesiadecki and R. D. Skeel, Dangers of multiple-time-step methods, J. Comput. Phys., 109 (1993), pp. 318--328.

....timesteps for V than for V hard . We might try the following time splitting integration method: Deltat=2, V Deltat, T V hard numerically N steps of Verlet with stepsize Deltat=N Deltat=2, V Such symplectic multiple time stepping methods were discovered independently at Illinois [6, 15] and Columbia [43] Nonsymplectic MTS methods go back over 25 years in the astrophysics literature [18] and 16 years in the MD literature [42] Typical nonsymplectic MTS methods will exhibit poor behavior on long enough time intervals. On the other hand, it seems that there may be serious accuracy ....

....years in the MD literature [42] Typical nonsymplectic MTS methods will exhibit poor behavior on long enough time intervals. On the other hand, it seems that there may be serious accuracy and stability problems with symplectic MTS methods. For example, the possibility of resonance is reported in [15, 6]. And there are other concerns that do not appear in the literature. Unless these can be cleared up, it seems more prudent to use the better nonsymplectic MTS methods proposed in [15, 6, 37] IMA, LeiReiSke, November 16, 1994 12 The use of MTS with a timestep fixed for each bonded interaction is ....

[Article contains additional citation context not shown here]

Biesiadecki, J. J. and R. D. Skeel, Dangers of multiple-time-step methods, J. Comput. Phys. 109, 318--328, 1993.

....timesteps for V than for V hard . We might try the following time splitting integration method: Deltat=2, V Deltat, T V hard numerically N steps of Verlet with stepsize Deltat=N Deltat=2, V Such symplectic multiple time stepping methods were discovered independently at Illinois [6, 15] and Columbia [43] Nonsymplectic MTS methods go back over 25 years in the astrophysics literature [18] and 16 years in the MD literature [42] Typical nonsymplectic MTS methods will exhibit poor behavior on long enough time intervals. On the other hand, it seems that there may be serious accuracy ....

....years in the MD literature [42] Typical nonsymplectic MTS methods will exhibit poor behavior on long enough time intervals. On the other hand, it seems that there may be serious accuracy and stability problems with symplectic MTS methods. For example, the possibility of resonance is reported in [15, 6]. And IMA, LeiReiSke, May 26, 1995 13 there are other concerns that do not appear in the literature. Unless these can be cleared up, it seems more prudent to use the better nonsymplectic MTS methods proposed in [15, 6, 37] The use of MTS with a timestep fixed for each bonded interaction is ....

[Article contains additional citation context not shown here]

Biesiadecki, J. J. and R. D. Skeel, Dangers of multiple-time-step methods, J. Comput. Phys. 109, 318--328, 1993.

No context found.

J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109(2):318-328, Dec. 1993.

No context found.

J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109(2):318-328, Dec. 1993.

No context found.

J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109(2):318--328, December 1993.

No context found.

J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109(2):318--328, 1993.

No context found.

J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109(2):318--328, Dec. 1993.

No context found.

Biesiadecki, J.J. and Skeel, R.D., Danger of multiple-time-step methods, J. Comput. Phys. 109, 318--328, 1993.

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