R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18(1):203-222, Jan. 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... that symplectic and time reversible discretization schemes possess particularly attractive properties when applied over a long time 1 to Hamiltonian systems [22, 12] Much attention has been paid recently in this context to the (implicit) midpoint scheme and some of its variants; see, e.g. [25, 10, 14] (see also [5] for a different orientation) The midpoint scheme, and more generally Gauss collocation schemes, are algebraically stable, symplectic [22] and they preserve quadratic integral invariants [8] They may appear particularly suitable for the numerical solution of highly oscillatory ....

R.D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18:203--222, 1997.

....K 2 = H p (p 1 ) 0 ; q n p n = q n Gamma1 p n Gamma1 k(K 1 K 2 ) q n Gamma1 p n Gamma1 k H p (p 1 ) GammaH q (q 1 ) We note that this scheme is explicit, and it can also be shown to be symplectic. It is identical to the leapfrog Verlet scheme [12] p n = p n Gamma1 Gamma kH q (q n Gamma1 k=2H p (p n Gamma1 ) q n = q n Gamma1 kH p p n p n Gamma1 2 : 2 IMEX RUNGE KUTTA SCHEMES 8 2.4 A third order combination (2,3,3) The two stage, third order DIRK scheme with the best damping properties turns out to be the DIRK scheme ....

Robert D. Skeel, Guihua Zhang, and Tamar Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18(1):203--222, 1997.

.... that symplectic and time reversible discretization schemes possess particularly attractive properties when applied over a long time 1 to Hamiltonian systems [22, 12] Much attention has been paid recently in this context to the (implicit) midpoint scheme and some of its variants; see, e.g. [25, 10, 14] (see also [5] for a different orientation) The midpoint scheme, and more generally Gauss collocation schemes, are algebraically stable, symplectic [22] and they preserve quadratic integral invariants [8] They may appear particularly suitable for the numerical solution of highly oscillatory ....

....better than the (cheaper, explicit) staggered midpoint scheme, but the error in J is smaller. For k = the errors in both total energy and in J are much smaller using the midpoint scheme, indicating its potential attraction despite the possible setbacks for ff not small. 2 4 The variant in [25] shifts the mesh by k=2. Acknowledgement: We thank Bob Skeel for many fruitful discussions that have led to significant improvements of our original manuscript. ....

R.D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18:203--222, 1997.

....pointwise accurate approximate solutions. But the time step restriction implies an enormous computational burden. Furthermore, in many cases the high frequency responses are of little or no interest. Consequently, various researchers have considered the use of semi implicit implicit methods, e.g. [6,11,9,15,18,12,13,8,17,3]. The work of this author was partially supported under NSERC Canada Grant OGP0004306. 2 Uri M. Ascher and Sebastian Reich A popular implicit discretization is the (implicit) midpoint method [7] which, applied to a system of the type d dt q = p ; 1a) d dt p = GammarU (q) 1b) yields ....

....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 multipletime stepping methods [4] or implicit methods [17,18,12,3]. 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 general class of semi implicit methods depicted in Fig. 1 [12] In this scheme, Step 3 of the nth time step can be combined with Step 1 ....

R.D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18:203--222, 1997.

.... that symplectic and time reversible discretization schemes possess particularly attractive properties when applied over a long time 1 to Hamiltonian systems [22, 12] Much attention has been paid recently in this context to the (implicit) midpoint scheme and some of its variants; see, e.g. [25, 10, 14] (see also [5] for a different orientation) The midpoint scheme, and more generally Gauss collocation schemes, are algebraically stable, symplectic [22] and they preserve quadratic integral invariants [8] They may appear particularly suitable for the numerical solution of highly oscillatory ....

....than the (cheaper, explicit) staggered midpoint scheme, but the error in J is smaller. For k = the errors in both total energy and in J are much smaller using the midpoint scheme, indicating its potential attraction despite the possible setbacks for ff not small. Upsilon 6 The variant in [25] shifts the mesh by k=2. 4. Conclusions. In this paper we have investigated the suitability of the midpoint scheme for highly oscillatory, frictionless mechanical systems, where the step size k is much larger than the system s small parameter , in case that the solution remains bounded as ....

R.D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18:203--222, 1997.

....pointwise accurate approximate solutions. But the time step restriction implies an enormous computational burden. Furthermore, in many cases the high frequency responses are of little or no interest. Consequently, various researchers have considered the use of semi implicit implicit methods, e.g. [6, 10, 9, 14, 16, 11, 12, 8, 15, 3]. A popular implicit discretization is the (implicit) midpoint method [7] which, applied to a system of the type d dt q = p ; 1a) d dt p = GammarU (q) 1b) yields the discretization q n 1 = q n k p n 1=2 ; 2a) p n 1 = p n Gamma k rU(q n 1=2 ) 2b) with p n 1=2 = p n 1 p n ] 2, etc. ....

....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 = p 1;n Gamma k V 0 1 (r 1;n ) 2r ....

R.D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18:203--222, 1997.

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R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18(1):203--222, Jan. 1997.

....so nonlinear as molecular dynamics. Numerical results are given at the end of this paper. 2 Mixed implicit explicit methods We begin by describing a family of simple implicit methods of conventional type for the system x = v; v = F (x) where F (x) GammaU x (x) and U(x) is potential energy [6]. At the beginning of a typical step we have available the collective positions X n and the collective velocities V n . We advance one step as follows: X n 1=2 = X n Deltat 2 V n ; F n 1=2 = F (X n 1=2 ff Deltat 2 F n 1=2 ) V n 1 = V n DeltatF n 1=2 ; X n 1 = X n 1=2 Deltat ....

R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18(1):203--222, January 1997.

.... 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 energy pulses are introduced to the governing dynamics equations when the slow forces are evaluated. These large pulses in turn lead to incorrect ....

....) cos sin Gamma sin cos G( Delta; Gamma1 to emphasize similarity of A V V to a rotation matrix, a consequence of symplecticness (conservation of area in phase space) The physical angular frequency Omega (eq. 2. 2) is numerically approximated by an effective angular frequency [14, 13] Omega eff = Delta; Delta = Omega O( Delta 3 ) 2.10) 2.2.2. Langevin Dynamics RESONANCE IN FORCE SPLITTING INTEGRATORS 9 We analyze the stability and resonance independent of the random force (i.e. R(t) 0) following [15, 11] We assume that the inner timestep is ....

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R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: Stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comp., 18:202--222, January 1997.

....compelling evidence against the use of floating point lattices; however, the existence of a simple and highly efficient fixed point lattice integration algorithm would strengthen the case for the use of fixed point lattices. In Section 2, we define a one parameter family of symplectic integrators [22] that includes leapfrog Stormer Verlet, Cowell Numerov, implicit midpoint trapezoid, and LIM2 [25] In Section 3, we devise a very efficient floatingpoint implementation of such a symplectic integrator, which requires only a simple change or two to the usual floating point version of the ....

R. D. Skeel, G. Zhang and T. Schlick, A family of symplectic integrators: stability, accuracy, and molecular dynamics applications, SIAM J. Sci. Comput. 18 (1997) 203--222.

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R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18(1):203-222, Jan. 1997.

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R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18(1):203-222, Jan. 1997.

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R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18(1):203--222, 1997.

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R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18(1):203-- 222, Jan. 1997.

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R.D. Skeel, G. Zhang, T. Schlick, A Family of Symplectic Integrators: Stability, Accuracy, and Molecular Dynamics Applications , SIAM J. Sci. Comput., Vol. 18, No. 1, 1997, pp. 203-222.

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R. D. Skeel, G. H. Zhang, and T. Schlick. A family of symplectic integrators: Stability, accuracy, and molecular dynamics applications. SIAM J. Scient. COMP., 18:203--222, 1997.

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R. D. Skeel, G. H. Zhang, and T. Schlick. A family of symplectic integrators: Stability, accuracy, and molecular dynamics applications. SIAM J. Scient. COMP., 18:203--222, 1997.

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