The Development of Variable-Step Symplectic Integrators, with Application to the Two-Body Problem

Calvo, M. P.; Sanz‐Serna, J. M.

doi:10.1137/0914057

Cited by 147 publications

(123 citation statements)

References 16 publications

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“…To this aim, the fourth-order explicit and symplectic Runge-Kutta-Nyström method developed in Ref. [30], with time step equal to 10 −3 , was used.…”

Section: Appendix B: Numerical Methods For Computing Solitary Travelimentioning

confidence: 99%

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

et al. 2017

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In this work, we provide two complementary perspectives for the (spectral) stability of solitary traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. One is as an eigenvalue problem for a stationary solution in a cotraveling frame, while the other is as a periodic orbit modulo shifts. We connect the eigenvalues of the former with the Floquet multipliers of the latter and using this formulation derive an energy-based spectral stability criterion. It states that a sufficient (but not necessary) condition for a change in the wave stability occurs when the functional dependence of the energy (Hamiltonian) H of the model on the wave velocity c changes its monotonicity. Moreover, near the critical velocity where the change of stability occurs, we provide an explicit leading-order computation of the unstable eigenvalues, based on the second derivative of the Hamiltonian H (c 0 ) evaluated at the critical velocity c 0 . We corroborate this conclusion with a series of analytically and numerically tractable examples and discuss its parallels with a recent energy-based criterion for the stability of discrete breathers.

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“…To this aim, the fourth-order explicit and symplectic Runge-Kutta-Nyström method developed in Ref. [30], with time step equal to 10 −3 , was used.…”

Section: Appendix B: Numerical Methods For Computing Solitary Travelimentioning

confidence: 99%

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

et al. 2017

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“…We emphasize that our purpose here is to illustrate the scope of symplectic integrators, rather than to offer definite conclusions as to the relative merits of symplectic methods when compared with their standard, nonsymplectic counterparts. The reader is referred to [3] for more serious numerical comparisons using variable-step software.…”

Section: A Numerical Illustrationmentioning

confidence: 99%

Partitioned Runge-Kutta Methods for Separable Hamiltonian Problems

Abia¹,

Sanz‐Serna²

1993

Mathematics of Computation

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Abstract. Separable Hamiltonian systems of differential equations have the form dp/dt = -dH/dq, dq/dt = dH/dp, with a Hamiltonian function H that satisfies H = T(p) + K(q) (T and V are respectively the kinetic and potential energies). We study the integration of these systems by means of partitioned Runge-Kutta methods, i.e., by means of methods where different Runge-Kutta tableaux are used for the p and q equations. We derive a sufficient and "almost" necessary condition for a partitioned Runge-Kutta method to be canonical, i.e., to conserve the symplectic structure of phase space, thereby reproducing the qualitative properties of the Hamiltonian dynamics. We show that the requirement of canonicity operates as a simplifying assumption for the study of the order conditions of the method.

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“…Since the appearance of the papers by Creutz and Gocksch [16], Yoshida [44], and Suzuki [40], considerable effort has been put into obtaining more efficient, explicit, constant time step symplectic integrators [30]. Particularly interesting are: symplectic partitioned Runge-Kutta (PRK) methods [28,3,6] for separable systems like H = T (p) + V (q); symplectic Runge-Kutta-Nyström (RKN) methods [14,28,5,6] for examples where T is quadratic in momenta; and near-integrable systems H = H 0 + H 1 , where both H 0 and H 1 are exactly solvable or easy to approximate, and is a small parameter [43,29,4]. Crucial to these methods is the separability of the Hamiltonian which allows the use of explicit methods.…”

Section: Symplectic Methods Given the Hamiltonian H(q P)mentioning

confidence: 99%

Adaptive Geometric Integrators for Hamiltonian Problems with Approximate Scale Invariance

Blanes¹,

Budd²

2005

SIAM J. Sci. Comput.

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Abstract. We consider adaptive geometric integrators for the numerical integration of Hamiltonian systems with greatly varying time scales. A time regularization is considered using either the Sundman or the Poincaré transformation. In the latter case, this gives a new Hamiltonian which is usually separable, but with one of the parts not always exactly solvable. This system can be numerically integrated with a splitting scheme where each part can be computed using a symplectic implicit or explicit method, preserving the qualitative properties of the exact solution. In this case, a backward error analysis for the numerical integration is presented. For a one-dimensional near singular problem, this analysis reveals a strong dependence of the performance of the method with the choice of the monitor function g, which is also observed when using other symmetric nonsymplectic integrators. We also show how this dependence greatly increases with the order of the numerical integrator used. The optimal choice corresponds to the function g, which nearly preserves the scaling invariance of the system. Numerical examples supporting this result are presented. In some cases a canonical transformation can also be considered, making the system more regular or easy to compute, and this is also illustrated with some examples. 1. Introduction. When solving Hamiltonian systems of ordinary differential equations, certain qualitative properties of the evolution are important, and symplectic integrators have largely shown during the last decade to be superior to standard integrators [36,10,20] when used with constant time step. In contrast, adaptive variable time step methods are often superior to fixed time step methods when applied to problems with varying evolutionary time scales. They lead to more regular problems with reduced local errors and with the effects of rounding error minimized. The results presented in the paper [8] demonstrate that adaptive methods can be especially effective when the underlying problem has a scaling structure. However, adaptive and symplectic methods have tended to sit uncomfortably together, with adaptivity often corrupting the powerful long time error estimates obtained for fixed time step symplectic methods [36]. Attempts to rectify this problem, which we will describe, have tended to result in either complex and hard to use algorithms or low order methods. It is also not clear in many of these methods what choice should be made of the adaptive procedure, and how this choice affects the performance of a numerical integrator of a given order. We present in this paper a technique for identifying a natural adaptive procedure based upon identifying the evolutionary scalings of the system, and then implementing this, using a combination of a Sundman and a Poincaré transform, for

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The Development of Variable-Step Symplectic Integrators, with Application to the Two-Body Problem

Cited by 147 publications

References 16 publications

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

Partitioned Runge-Kutta Methods for Separable Hamiltonian Problems

Adaptive Geometric Integrators for Hamiltonian Problems with Approximate Scale Invariance

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