Simulation of one-dimensional noisy Hamiltonian systems and their application to particle storage rings

Seeßelberg, M.; Breuer, Heinz‐Peter; Mais, H.; Petruccione, Francesco; Honerkamp, Josef

doi:10.1007/bf01559525

Cited by 17 publications

(17 citation statements)

References 4 publications

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“…Some physical applications of stochastic symplectic integration for such systems are discussed in [14]. Due to specific features of the system (4.2), (4.3) we have succeeded in construction of explicit partitioned Runge-Kutta (PRK) methods of a higher order.…”

Section: Explicit Symplectic Methods In the Case Of Separable Hamiltomentioning

confidence: 99%

“…In [14] a model describing synchrotron oscillations of particles in storage rings under the influence of external fluctuating electromagnetic fields was considered. This model can be written in the following form: dP = −ω 2 sin(Q)dt − σ 1 cos(Q)dw 1 − σ 2 sin(Q)dw 2 , (5.5) dQ = P dt.…”

Section: A Model For Synchrotron Oscillations Of Particles In Storagementioning

confidence: 99%

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Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Milstein¹,

Repin²,

Tretyakov³

2002

SIAM J. Numer. Anal.

163

227

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Abstract. Stochastic Hamiltonian systems with multiplicative noise, phase flows of which preserve symplectic structure, are considered. To construct symplectic methods for such systems, sufficiently general fully implicit schemes, i.e., schemes with implicitness both in deterministic and stochastic terms, are needed. A new class of fully implicit methods for stochastic systems is proposed. Increments of Wiener processes in these fully implicit schemes are substituted by some truncated random variables. A number of symplectic integrators is constructed. Special attention is paid to systems with separable Hamiltonians. Some results of numerical experiments are presented. They demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones.

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Section: Explicit Symplectic Methods In the Case Of Separable Hamiltomentioning

confidence: 99%

Section: A Model For Synchrotron Oscillations Of Particles In Storagementioning

confidence: 99%

Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Milstein¹,

Repin²,

Tretyakov³

2002

SIAM J. Numer. Anal.

163

227

View full text Add to dashboard Cite

show abstract

“…In [14] a symplectic method of mean-square order 1 for (5.2)-(5.3) is proposed on the basis of the Störmer-Verlet method. There, some physical applications of stochastic symplectic integrators are also discussed.…”

Section: Explicit Methods Of Ordermentioning

confidence: 99%

Symplectic Integration of Hamiltonian Systems with Additive Noise

Milstein¹,

Repin²,

Tretyakov³

2002

SIAM J. Numer. Anal.

140

155

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Abstract. Hamiltonian systems with additive noise possess the property of preserving symplectic structure. Numerical methods with the same property are constructed for such systems. Special attention is paid to systems with separable Hamiltonians and to second-order differential equations with additive noise. Some numerical tests are presented. Key words.Hamiltonian systems with additive noise, symplectic integration, mean-square methods for stochastic differential equations

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“…. Consider a stochastically perturbed Hamiltonian system of a pendulum ( [28], where we set λ A = 0) with Hamiltonians…”

Section: Example 3 (Synchrotron Oscillations)mentioning

confidence: 99%

General order conditions for stochastic partitioned Runge–Kutta methods

2017

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Abstract. In this paper stochastic partitioned Runge-Kutta (SPRK) methods are considered. A general order theory for SPRK methods based on stochastic B-series and multicolored, multishaped rooted trees is developed. The theory is applied to prove the order of some known methods, and it is shown how the number of order conditions can be reduced in some special cases, especially that the conditions for preserving quadratic invariants can be used as simplifying assumptions.

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Simulation of one-dimensional noisy Hamiltonian systems and their application to particle storage rings

Cited by 17 publications

References 4 publications

Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Symplectic Integration of Hamiltonian Systems with Additive Noise

General order conditions for stochastic partitioned Runge–Kutta methods

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