Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems

Cano, Begoña

doi:10.1093/imanum/18.1.57

Cited by 25 publications

(43 citation statements)

References 13 publications

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“…Remark 3.5. Notice that the results given by Theorem 3.4 for variable stepsizes are the same that were already proved in [7] for fixed stepsizes.…”

Section: Theorem 32 With the Preceding Notationsupporting

confidence: 72%

“…Theorem 2.2 says that b r1 (t) = −c r y (r+1) (t), and the result follows in the same way as in the proof for fixed stepsizes and even indexes [7].…”

Section: Theorem 32 With the Preceding Notationmentioning

confidence: 58%

“…The proof is similar to that of Lemma 1 in [7], where b j1 was substituted by its fixed-stepsize counterpart −c j y (j+1) , and follows from the periodicity of b j1 (Remark 2.4) and the properties of transition matrices.…”

Section: Error Growth In the Integration Of Periodic Orbitsmentioning

confidence: 78%

“…Let us also assume that the solution to (1.1) is symmetric, i.e., the corresponding trajectory in phase space intersects the invariant subspace X + of Λ. In such a case, the monodromy matrix of the orbit has special properties which allow a favourable growth of the coefficients e j1 (even j) of the global error for every FSLMM [7]. For variable stepsizes, however, it is only possible to assure that good behaviour for the first term e r1 (t) and just in the case r is even.…”

Section: Theorem 32 With the Preceding Notationmentioning

confidence: 99%

“…On the other hand, for fixed-stepsize linear multistep methods (FSLMM), an analysis has been given in [7] concerning the growth of error with time when integrating periodic orbits. For reversible systems, symmetric linear multistep methods for first-order differential systems were not recommended in general because they lead to exponential error growth.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

Cano

Durán

2003

Math. Comp.

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Abstract. In this paper we deal with several issues concerning variablestepsize linear multistep methods. First, we prove their stability when their fixed-stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients. Then we prove asymptotic expansions on the considered tolerance for the global error committed. Using them, we study the growth of error with time when integrating periodic orbits. We consider strongly and weakly stable linear multistep methods for the integration of first-order differential systems as well as those designed to integrate special second-order ones. We place special emphasis on the latter which are also symmetric because of their suitability when integrating moderately eccentric orbits of reversible systems. For these types of methods, we give a characterization for symmetry of the coefficients, which allows their construction, and provide some numerical results for them.

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“…Remark 3.5. Notice that the results given by Theorem 3.4 for variable stepsizes are the same that were already proved in [7] for fixed stepsizes.…”

Section: Theorem 32 With the Preceding Notationsupporting

confidence: 72%

“…Theorem 2.2 says that b r1 (t) = −c r y (r+1) (t), and the result follows in the same way as in the proof for fixed stepsizes and even indexes [7].…”

Section: Theorem 32 With the Preceding Notationmentioning

confidence: 58%

Section: Error Growth In the Integration Of Periodic Orbitsmentioning

confidence: 78%

Section: Theorem 32 With the Preceding Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

Cano

Durán

2003

Math. Comp.

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Variable Stepsizes in Symmetric Linear Multistep Methods

Cano

2001

Lecture Notes in Computer Science

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A Brief Introduction to Variational Integrators

Lew

2016

Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics

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In this chapter, a brief introduction to the formulation of variational methods for finite-dimensional Lagrangian systems is presented. To this end, the first two sections focus on describing the Lagrangian and Hamiltonian points of view of mechanics for systems evolving on manifolds. Special attention is paid to the construction of the Lagrangian function and to the role of Hamilton's variational principle in the deduction of the balance equations. The relation between the symmetries of the Lagrangian function and the existence of invariants of the dynamics along with the symplectic nature of the flow are also addressed. In the third section, the discussion turns towards the formulation of a time-discrete analogue of the theory. The cornerstone of such a construction is given by a discrete analogue of Hamilton's variational principle which provides a systematic procedure to construct discrete approximations to the exact trajectory of a mechanical system on both the configuration space and the phase space. The approximation properties and the geometric characteristics of the resulting discrete trajectories are explained. Finally, we apply the variational methodology to construct symplectic and momentum-conserving time integrators for two problems of practical interest in engineering and science.

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Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems

Cited by 25 publications

References 13 publications

Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

Variable Stepsizes in Symmetric Linear Multistep Methods

A Brief Introduction to Variational Integrators

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