ERKN integrators for systems of oscillatory second-order differential equations

Wu, Xinyuan; You, Xiong; Shi, Wei; Wang, Bin

doi:10.1016/j.cpc.2010.07.046

Cited by 109 publications

(35 citation statements)

References 17 publications

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“…Taking account of the oscillatory structure generated by the linear part My, Wu et al [24] proposed a standard form of the multi-frequency and multidimensional ERKN methods for (23) and derived the corresponding order conditions based on the extended Nyström tree theory, a tri-colored rooted tree theory (readers are referred to [24,25]). A remarkable advantage of ERKN methods is that both their internal stages and updates take into account the special oscillatory structures of the system (23) introduced by the linear term My, therefore, they can naturally integrate the multi-frequency and multidimensional oscillatory homogeneous system y 00 ðtÞ þ MyðtÞ ¼ 0 exactly.…”

Section: Multidimensional Extended Runge-kutta-nyström Integratorsmentioning

confidence: 99%

An efficient high-order explicit scheme for solving Hamiltonian nonlinear wave equations

Liu

Shi

2014

Applied Mathematics and Computation

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Section: Multidimensional Extended Runge-kutta-nyström Integratorsmentioning

confidence: 99%

An efficient high-order explicit scheme for solving Hamiltonian nonlinear wave equations

Liu

Shi

2014

Applied Mathematics and Computation

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“…Finally, the extension of the new methods proposed in this paper to the case where the ω 2 in (1) is a positive semi-definite is direct just in an analogous way to Wu et al [6] for multi-dimensional ARKN methods or Wu et al [19] for multi-dimensional ERKN methods.…”

Section: Discussionmentioning

confidence: 99%

Trigonometrically-fitted Scheifele two-step methods for perturbed oscillators

You

Zhang

Zhao

2011

Computer Physics Communications

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“…We refer the reader to [5,6,9,12,23,24] for example. Recently, Wu et al [23] took account of the special structure of the system (5.6) introduced by the linear term Mq, which leads to the high-frequency oscillation in the solution, and formulated a standard form for a multi-frequency ERKN (extended RKN) integrator. ERKN integrators exhibit the correct qualitative behaviour much better than classical RKN methods due to using the special structure of the equation introduced by the linear term Mq.…”

Section: It Is Known That the Hamiltonian H(y) = H(p Q)mentioning

confidence: 99%

An Extended Discrete Gradient Formula for Multi-frequency Oscillatory Hamiltonian Systems

Shi

2015

Structure-Preserving Algorithms for Oscillatory Differential Equations II

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The purpose of this chapter is to introduce an extended discrete gradient formula for the multi-frequency oscillatory Hamiltonian system with the Hamiltonianrepresents generalized momenta, and M ∈ R d×d is a symmetric and positive semi-definite matrix. The solution of this system is a nonlinear oscillator. Basically, many nonlinear oscillatory mechanical systems with a partitioned Hamiltonian function lend themselves to this approach. The extended discrete gradient formula exactly preserves the energy H(p, q). Some properties of the new formula are given. The convergence is analysed for the implicit schemes based on the discrete gradient formula, and it turns out that the convergence is independent of M , which is a significant property for numerically solving the oscillatory Hamiltonian system. This implies that a larger stepsize can be chosen for the extended energypreserving scheme than for the traditional discrete gradient methods in applications to multi-frequency oscillatory Hamiltonian systems. Illustrative examples show that the new schemes are greatly superior to the traditional discrete gradient methods in the literature. MotivationThe design and analysis of numerical integration methods for nonlinear oscillators is an important problem that has received a great deal of attention in the last few years. It is known that the traditional approach to deriving numerical integration methods is based on the natural procedure of discretizing the differential equation in such a way as to make the local truncation errors associated with the discretization as small as possible. A relatively new and increasingly important area in numerical integration methods is geometric integration. A numerical integration method is called geometric if it exactly preserves one or more physical/geometric properties

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ERKN integrators for systems of oscillatory second-order differential equations

Cited by 109 publications

References 17 publications

An efficient high-order explicit scheme for solving Hamiltonian nonlinear wave equations

An efficient high-order explicit scheme for solving Hamiltonian nonlinear wave equations

Trigonometrically-fitted Scheifele two-step methods for perturbed oscillators

An Extended Discrete Gradient Formula for Multi-frequency Oscillatory Hamiltonian Systems

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