Symplectic method

E. Hairer. Backward analysis of numerical integrators and symplectic methods. Annals of Numerical Mathematics 1 (1994)... 

The IE and IM methods described above turn out to be quite special in that IE s damping is extreme and IM s resonance patterns are quite severe relative to related symplectic methods. However, success was not much greater with a symplectic implicit Runge-Kutta integrator examined by Janezic and coworkers [40],... 

D. Janezic and F. Merzel. Split integration symplectic method for molecular dynamics integration. J. Chem. Inf. Comput. Sci., 37 1048-1054, 1997. 

Long Time Step MD Simulations Using Split Integration Symplectic Method ... 

Fig. 1. The Split Integration Symplectic Method (SISM) solution procedure.

Note that there are also variations in total energy which might be due to the so called step size resonance [26, 27]. Shown are also results for fourth order algorithm which gives qualitatively the same results as the second order SISM. This show that the step size resonances are not due to the low order integration method but rather to the symplectic methods [28]. 

Verlet, L. Computer Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review 159 (1967) 98-103 Janezic, D., Merzel, F. Split Integration Symplectic Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 37 (1997) 1048-1054 McLachlan, R. I. On the Numerical Integration of Ordinary Differential Equations by Symplectic Composition Methods. SIAM J. Sci. Comput. 16 (1995) 151-168... 

Janezic, D., Merzel, F. An Efficient Split Integration Symplectic Method for Molecular Dynamics Simulations of Complex Systems. In Proceedings of the... 

In this article, we briefly describe these symplectic methods, citing recent articles for most of the details of derivation and implementation. We compare the various algorithms in terms of theoretical and implementation aspects, as well as in simple numerical experiments. 

The concept of a symplectic method is easily extended to systems subject to holonomic constraints [22]. For example the RATTLE discretization is found to be a symplectic discretization. Since SHAKE is algebraically equiva lent to RATTLE, it, too, has the long-term stability of a symplectic method. 

In experiments, the two symplectic methods ROT and SPL performed very similarly in terms of error propagation and long term stability. The ex-... 

Note the curious return property exhibited by the energy in the symplectic method this is a manifestation of the nearby Hamiltonian mentioned in the introduction (see [7, 18, 28]). 

These experiments confirm observations in the recent articles [20] and [11] symplectic methods easily outperform more traditional quaternionic integration methods in long term rigid body simulations. 

Long term simulations require structurally stable integrators. Symplec-tic and symmetric methods nearly perfectly reproduce structural properties of the QCMD equations, as, for example, the conservation of the total energy. We introduced an explicit symplectic method for the QCMD model — the Pickaback scheme— and a symmetric method based on multiple time stepping. 

We focus on so-called symplectic methods [18] for the following reason It has been shown that the preservation of the symplectic structure of phase space under a numerical integration scheme implies a number of very desirable properties. Namely,... 

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... 

The authors in this paper present an explicit symplectic method for the numerical solution of the Schrodinger equation. A modified symplectic integrator with the trigonometrically fitted property which is based on this method is also produced. Our new methods are tested on the computation of the eigenvalues of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator and the Morse potential. 

T. Mono Vasilis, Z. Kalogiratou and T. E. Simos, Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrodinger equation,... 

K. Tselios and T. E. Simos, Symplectic methods of fifth order for the numerical solution of the radial Shrodinger equation, J. Math. Chem., 2004, 35(1), 55-63. 

Th. Monovasilis and Z. Kalogiratou, Trigonometrically and Exponentially fitted Symplectic Methods of third order for the Numerical Integration of the Schrodinger Equation, Appl. Num. Anal. Comp. Math., 2005, 2(2), 238-244. 

The behavior of the Runge-Kutta-Nystrom Symplectic method of algebraic order four developed by Sanz-Serna and Calvo12 and the behavior of the classical partitioned multistep method is similar. These methods are much more efficient that the embedded Runge-Kutta method of Dormand and Prince 5(4) (see 13). 

In 32 the authors consider the solution of the two-dimensional time-independent Schrodinger equation by partial discretization. The discretized problem is treated as a problem of the numerical solution of system of ordinary differential equations and has been solved numerically by symplectic methods. The problem is then transformed into an algebraic eigenvalue problem involving real, symmetric, large sparse matrices. As numerical illustrations the authors have found the eigenvalues of the two-dimensional harmonic oscillator... 

In 34 the eigenvalue problem of the one-dimensional time-independent Schrodinger equation is studied. Exponentially fitted and trigonometrically fitted symplectic integrators are developed, by modification of the first and second order Yoshida symplectic methods. Numerical results are presented for the one-dimensional harmonic oscillator and Morse potential. 

This is an exact transcription of the formulas in Newmark s original paper, only substituting M p for the velocity v wherever it appears. In practice the choice a = 1/2 is used to avoid spurious damping (it can be demonstrated for a simple model problem) this certainly would appear to be desirable in the setting of molecular dynamics. For rj = 0 we then arrive at the Verlet method. For other values of r the scheme is clearly implicit, which likely is the reason it is rarely used in molecular simulation, although it is popular in structural mechanics. The implicit Newmark methods are not symplectic, but a related family of symplectic methods can be constructed by replacing interpolated forces by forces evaluated at interpolated positions [395]. 

Euler s method is not symplectic for a general Hamiltonian system. Similarly for a general divergence free vector field, Euler s method is not volume preserving. Find conditions on the vector field that imply that Euler s method is volume preserving. Are there special Hamiltonian systems for which Euler s method is a symplectic method ... 

Is it true that any symplectic method will have a symplectic adjoint method Either give a proof or find a counterexample. 

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