Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs

Hu, Weipeng; Deng, Zichen; Han, Songmei; Zhang, Wenrong

doi:10.1016/j.jcp.2012.10.032

Cited by 122 publications

(57 citation statements)

References 13 publications

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“…According to the deduction process of the multisymplectic form, with the canonical momenta @ x v ¼ u, w ¼ @ x u, @ x p ¼ À@ t u=2 and the state variable z ¼ ½v, u, w, p T 2 R 4 , equation (1) can be written as the following generalized multi-symplectic form (Hu et al, 2013a), which is very similar to the multi-symplectic form (2): Equation (3) is known as the generalized multi-symplectic form because it satisfies the following generalized multi-symplectic conservation law (Hu et al, 2013a) with small damping coefficient:…”

Section: Generalized Multi-symplectic Form Of Kdv-burgers Equationmentioning

confidence: 99%

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Competition between geometric dispersion and viscous dissipation in wave propagation of KdV-Burgers equation

Deng

2014

Journal of Vibration and Control

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In this paper the competitive relationship between the geometric dispersion and the viscous dissipation in the wave propagation of the KdV-Burgers equation is investigated by the generalized multi-symplectic method. Firstly, the generalized multi-symplectic formulations for the KdV-Burgers equation are presented in Hamiltonian space. Then, focusing on the inherent geometric properties of the generalized multi-symplectic formulations, a 12-point difference scheme is constructed. Finally, numerical experiments are performed with fixed step-sizes to obtain the maximum damping coefficient that insures that the scheme constructed is generalized multi-symplectic, and to study the competition between the geometric dispersion and the viscous dissipation in the wave propagation of the KdV-Burgers equation. The competition phenomena are comprehensively illustrated in the wave forms as well as in the phase diagrams: for the KdV equation (a particular case of the KdV-Burgers equation without dissipation), there is a closed orbit in the phase diagram; and the closed orbit is substituted by a heteroclinic one with the appearance of the viscous dissipation; moreover, the heteroclinic orbit changes from the saddle-node type to the saddle-focus type with an increase of the damping coefficient.

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Section: Generalized Multi-symplectic Form Of Kdv-burgers Equationmentioning

confidence: 99%

“…Here, the right hand item of equation (4) is defined as the error of the generalized multi-symplectic conservation law Á ¼ À@ x ðdzÃKdzÞ (Hu et al, 2013a), the discrete value of which will be used to evaluate whether the numerical scheme constructed is generalized multi-symplectic or not.…”

Section: Generalized Multi-symplectic Form Of Kdv-burgers Equationmentioning

confidence: 99%

Competition between geometric dispersion and viscous dissipation in wave propagation of KdV-Burgers equation

Deng

2014

Journal of Vibration and Control

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“…Many physical problems are designed in mathematical form using nonlinear PDEs [1][2][3]. The generalized Burgers-Fisher (GBF) and generalized Burgers-Huxley (GBH) equations describe many physical phenomena.…”

Section: Introductionmentioning

confidence: 99%

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

Wasim

Abbas

Noor‐ul‐Amin

2018

Mathematical Problems in Engineering

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In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier) method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature.

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“…Moreover, if damping of the structure is considered, the dissipative effect cannot be simulated by these numerical methods. The generalized multi-symplectic method [Hu et al, 2012a;Hu et al, 2013a;Hu et al, 2013b;Hu et al, 2012b] derived from the multi-symplectic idea [Bridges, 1997;Bridges and Reich, 2001;Moore and Reich, 2003a;Moore and Reich, 2003b;Reich, 2000], proved to be a high-efficiency structure-preserving method for analyzing the moving load system and the dissipative problem [Hu et al, 2012a;Hu et al, 2012b], is now used to study the dynamic responses of a nonuniform continuous beam with small damping coefficient under moving loads in this paper. The results of the numerical experiments are presented for the beam under a single load moving at a constant speed or a variable speed.…”

Section: Introductionmentioning

confidence: 99%

Generalized Multi-Symplectic Method for Dynamic Responses of Continuous Beam Under Moving Load

Deng

Ouyang

2013

Int. J. Appl. Mechanics

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Based on the multi-symplectic idea, a generalized multi-symplectic integrator method is presented to analyze the dynamic response of multi-span continuous beams with small damping coefficient. Focusing on the local conservation properties, the generalized multisymplectic formulations are introduced and a fifteen-point implicit structure-preserving scheme is constructed to solve the first-order partial differential equations derived from the dynamic equation governing the dynamic behavior of continuous beams under moving load. From the results of the numerical experiments, it can be concluded that, for the cases considered in this paper, the structure-preserving scheme is generalized multisymplectic if the viscous damping c ≤ 0.3751 when the continuous beam is under a constant-speed moving load and the structure-preserving scheme is generalized multisymplectic if the viscous damping c ≤ 0.3095 when the continuous beam is under a variable-speed moving load with fixed step lengths ∆t = 0.05 and ∆x = 0.025. Similar to a multi-symplectic scheme, the generalized multi-symplectic scheme also has two remarkable advantages: the excellent long-time numerical behavior and the good conservation property.

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Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs

Cited by 122 publications

References 13 publications

Competition between geometric dispersion and viscous dissipation in wave propagation of KdV-Burgers equation

Competition between geometric dispersion and viscous dissipation in wave propagation of KdV-Burgers equation

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

Generalized Multi-Symplectic Method for Dynamic Responses of Continuous Beam Under Moving Load

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