General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs

Li, Yuwen; Wu, Xinyuan

doi:10.1016/j.jcp.2015.08.023

Cited by 39 publications

(21 citation statements)

References 31 publications

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“…In recent years, the local structurepreserving method has been of high interest in studying Hamiltonian PDEs (e.g., see Refs. [6,15,26] and references therein). However, to the best of our knowledge, there has been no reference considering a linearly implicit scheme for energy-conserving systems, which can preserve the local energy conservation law.…”

Section: Introductionmentioning

confidence: 99%

A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach

2019

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In this paper, we develop a novel, linearly implicit and local energy-preserving scheme for the sine-Gordon equation. The basic idea is from the invariant energy quadratization approach to construct energy stable schemes for gradient systems, which are energy dispassion. We here take the sine-Gordon equation as an example to show that the invariant energy quadratization approach is also an efficient way to construct linearly implicit and local energy-conserving schemes for energyconserving systems. Utilizing the invariant energy quadratization approach, the sine-Gordon equation is first reformulated into an equivalent system, which inherits a modified local energy conservation law. The new system are then discretized by the conventional finite difference method and a semi-discretized system is obtained, which can conserve the semi-discretized local energy conservation law. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semidiscrete system to arrive at a fully discretized scheme. We prove that the resulting scheme can exactly preserve the discrete local energy conservation law. Moveover, with the aid of the classical energy method, an unconditional and optimal error estimate for the scheme is established in discrete H 1 h -norm. Finally, various numerical examples are addressed to confirm our theoretical analysis and demonstrate the advantage of the new scheme over some existing local structure-preserving schemes.

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Section: Introductionmentioning

confidence: 99%

A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach

2019

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“…With (41), the defect (39) has the bound δ(t) s ≤ Cǫ N +1 for t ≤ ǫ −1 . For the defect in the initial conditions (23) and (24), it holds that…”

Section: Defectsmentioning

confidence: 99%

“…EP methods can exactly preserve the energy of the considered system. With regard to some examples of this topic, we refer the reader to [2,6,23,25,26,28,32]. Unfortunately however, it seems that the long-time behaviour of EP methods in other structure-preserving aspects has not been studied for wave equations in the literature, such as the numerical conservation of momentum and actions.…”

Section: Introductionmentioning

confidence: 99%

Long-time momentum and actions behaviour of energy-preserving methods for semi-linear wave equations via spatial spectral semi-discretisations

Wang

2019

Adv Comput Math

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As is known that wave equations have physically very important properties which should be respected by numerical schemes in order to predict correctly the solution over a long time period. In this paper, the long-time behaviour of momentum and actions for energy-preserving methods is analysed for semilinear wave equations. A full discretisation of wave equations is derived and analysed by firstly using a spectral semi-discretisation in space and then by applying the adopted average vector field (AAVF) method in time. This numerical scheme can exactly preserve the energy of the semi-discrete system. The main theme of this paper is to analyse another important physical property of the scheme. It is shown that this scheme yields near conservation of a modified momentum and modified actions over long times. Both the results are rigorously proved based on the technique of modulated Fourier expansions in two stages. First a multi-frequency modulated Fourier expansion of the AAVF method is constructed and then two almost-invariants of the modulation system are derived.

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“…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%

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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General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs

Cited by 39 publications

References 31 publications

A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach

A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach

Long-time momentum and actions behaviour of energy-preserving methods for semi-linear wave equations via spatial spectral semi-discretisations

Linearly implicit structure-preserving schemes for Hamiltonian systems

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