Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs

Gong, Yuezheng; Cai, Jiaxin; Wang, Yushun

doi:10.1016/j.jcp.2014.09.001

Cited by 84 publications

(67 citation statements)

References 19 publications

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“…which is preserved by the fully implicit global energy-preserving scheme of [16]. The proof of the following proposition is similar to the proof of Proposition 1, and hence omitted.…”

Section: (422)mentioning

confidence: 82%

“…In [16], Gong, Cai and Wang present a scheme that preserves the local energy conservation law (3.2) of a one-dimensional multi-symplectic PDE, obtained by applying the midpoint rule in space and the averaged vetor field (AVF) method in time. They also present schemes that preserve the global energy, but not (3.2), obtained by considering spatial discretizations that preserve the skew-symmetric property of the difference operator ∂ x .…”

Section: New Linearly Implicit Energy-preserving Schemesmentioning

confidence: 99%

“…Note that (4.11) is the discrete energy preserved by the fully implicit local energy-preserving method of [16]. Also, for methods based on the multi-symplectic structure, instead of solving for z directly, the normal procedure is to eliminate the auxiliary variables from the scheme and get an equation for one variable u.…”

Section: A Local Energy-preserving Scheme For Multi-symplectic Pdesmentioning

confidence: 99%

“…The concept of a multi-symplectic structure for PDEs was introduced by Bridges in [10,11], see also [14] for a framework based on a Lagrangian formulation of the Cartan form. Local energy-preserving methods were first studied in [15], and have garnered much interest recently, see for example [13,16,17].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Linearly implicit structure-preserving schemes for Hamiltonian systems

Eidnes¹,

Li²,

Sato

2021

Journal of Computational and Applied Mathematics

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We present linearly implicit methods that preserve discrete approximations to local and global energy conservation laws for multi-symplectic PDEs with cubic invariants. The methods are tested on the one-dimensional Korteweg-de Vries equation and the two-dimensional Zakharov-Kuznetsov equation; the numerical simulations confirm the conservative properties of the methods, and demonstrate their good stability properties and superior running speed when compared to fully implicit schemes.

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“…which is preserved by the fully implicit global energy-preserving scheme of [16]. The proof of the following proposition is similar to the proof of Proposition 1, and hence omitted.…”

Section: (422)mentioning

confidence: 82%

Section: New Linearly Implicit Energy-preserving Schemesmentioning

confidence: 99%

Section: A Local Energy-preserving Scheme For Multi-symplectic Pdesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Linearly implicit structure-preserving schemes for Hamiltonian systems

Eidnes¹,

Li²,

Sato

2021

Journal of Computational and Applied Mathematics

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show abstract

“…Actually as the boundary conditions are periodic or homogeneous, the local energy-preserving/momentum-preserving schemes will be global energy-preserving/momentumpreserving schemes. On construction of the local structure-preserving schemes for some classical PDEs, we have done some works [15][16][17]. But there is no work on convergence analysis of these proposed local structure-preserving schemes.…”

Section: Introductionmentioning

confidence: 99%

Local energy‐ and momentum‐preserving schemes for Klein‐Gordon‐Schrödinger equations and convergence analysis

Cai

Hong²,

Wang

2017

Numerical Methods Partial

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In this article, we obtain local energy and momentum conservation laws for the Klein-Gordon-Schrödinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy-and momentum-preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time-space region. With suitable boundary conditions, the schemes will be charge-and energy-/momentumpreserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order O(τ 2 + h 2 ). The theoretical properties are verified by numerical experiments.

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Numeric‐analytical solution of an LC circuit with nonlinear capacitor based on the symplectic conservative perturbation method

Yang

Wang

et al. 2021

Int J Numerical Modelling

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The LC circuit with a nonlinear capacitor is analyzed using the symplectic conservative perturbation method based on the symplectic matrix. By treating the electric charge and the flux linkage as dual variables, the LC circuit system can be transformed into a Hamiltonian system. Then we solve the nonlinear Hamiltonian matrix equation by the perturbation method. The Hamiltonian matrix can be cast into linear and nonlinear parts. Based on the exact solution of the linear part, the nonlinear part is solved perturbatively by a canonical transformation. Since the coefficient matrix of the obtained equation is still a Hamiltonian matrix, symplectic conservation is guaranteed. The validity of the method is demonstrated, and the stability, precision, and the efficiency of the method are discussed by comparing it with the fourth‐order Runge–Kutta method (RK4), the ordinary perturbation method, the small‐parameter perturbation method based on displacement (SPPD), and the harmonic balance (HB) method. Numerical results show that the proposed method has better stability, precision, and efficiency compared with the other methods and thus has great advantages in long‐term simulations.

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Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs

Cited by 84 publications

References 19 publications

Linearly implicit structure-preserving schemes for Hamiltonian systems

Linearly implicit structure-preserving schemes for Hamiltonian systems

Local energy‐ and momentum‐preserving schemes for Klein‐Gordon‐Schrödinger equations and convergence analysis

Numeric‐analytical solution of an LC circuit with nonlinear capacitor based on the symplectic conservative perturbation method

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