Symplectic Schemes for Birkhoffian System

Su, Hongling; Qin, Meng

doi:10.1088/0253-6102/41/3/329

Cited by 20 publications

(13 citation statements)

References 13 publications

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“…Given appropriate boundary conditions such as E ×⃗ n = 0 or H ×⃗ n or periodic boundary conditions, by direct integrating (25), (27) and (28), respectively, we obtain the following global conservation laws:…”

Section: Autonomous Birkhoffian Multi-symplectic Formulationmentioning

confidence: 99%

“…The above scheme also can be derived by applying the central box scheme directly to the Hamiltonian formulation (33). to [26,27,28]. The generating functional method is a new developing theory proposed recently and needs more deep discussions.…”

Section: Central Box Scheme In Hamiltonian Formulationmentioning

confidence: 99%

“…By the proposition 3.3.1, the conservation laws (8), (14), (17) and (19) can be transformed respectively into (27), (28), (37) and (39). Therefore, using the result in Proposition 4.1.1, we come up the following conclusions.…”

Section: Formal Energy/momentum Conservationmentioning

confidence: 99%

“…When σ 1 = σ 2 , it also preserves exactly the quadratic conservation laws (8) and (14). Here we give the discrete version of the local conservation laws (25), (27) and (28). The discrete version of (25) is…”

Section: Formal Energy/momentum Conservationmentioning

confidence: 99%

“…and the proposition 4.0.2, we obtain the following conclusions for the difference system (70): The scheme (70) preserves the discrete versions of multi-symplectic energy and momentum conservation laws (37) and (39). If σ 1 = σ 2 , it also preserves the discrete versions of the quadratic conservation law(27) and(28).…”

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confidence: 99%

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Energy/dissipation-preserving Birkhoffian multi-symplectic methods for Maxwell's equations with dissipation terms

2016

Journal of Computational Physics

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In this paper, we propose two new energy/dissipation-preserving Birkhoffian multi-symplectic methods (Birkhoffian and Birkhoffian box) for Maxwell's equations with dissipation terms. After investigating the non-autonomous and autonomous Birkhoffian formalism for Maxwell's equations with dissipation terms, we first apply a novel generating functional theory to the non-autonomous Birkhoffian formalism to propose our Birkhoffian scheme, and then implement a central box method to the autonomous Birkhoffian formalism to derive the Birkhoffian box scheme. We have obtained four formal local conservation laws and three formal energy global conservation laws. We have also proved that both of our derived schemes preserve the discrete version of the global/local conservation laws. Furthermore, the stability, dissipation and dispersion relations are also investigated for the schemes. Theoretical analysis shows that the schemes are unconditionally stable, dissipation-preserving for Maxwell's equations in a perfectly matched layer (PML) medium and have second order accuracy in both time and space. Numerical experiments for problems with exact theoretical results are given to demonstrate that the Birkhoffian multi-symplectic schemes are much more accurate in preserving energy than both the exponential finite-difference time-domain (FDTD) method and traditional Hamiltonian scheme. We also solve the electromagnetic pulse (EMP) propagation problem and the numerical results show that the Birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation.

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Section: Autonomous Birkhoffian Multi-symplectic Formulationmentioning

confidence: 99%

Section: Central Box Scheme In Hamiltonian Formulationmentioning

confidence: 99%

Section: Formal Energy/momentum Conservationmentioning

confidence: 99%

Section: Formal Energy/momentum Conservationmentioning

confidence: 99%

mentioning

confidence: 99%

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Energy/dissipation-preserving Birkhoffian multi-symplectic methods for Maxwell's equations with dissipation terms

2016

Journal of Computational Physics

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Dynamic Symmetry Breaking and Generalized Multi-symplectic Method for Non-conservative Systems

Xiao

Deng

2023

Geometric Mechanics and Its Applications

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Fractional Birkhoffian mechanics

Shao-Kai

2014

Acta Mech

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In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Birkhoffian system with fractional derivatives (the fractional Birkhoffian mechanics), which gives a general method for constructing a fractional dynamical model of the actual problem. By using the definition of combined fractional derivative, we present a unified fractional Pfaff action and a unified fractional Pfaff-Birkhoff principle, and give four kinds of fractional Pfaff-Birkhoff principles under the different definitions of fractional derivative. And then, by using the fractional Pfaff-Birkhoff principles, we establish a series of fractional Birkhoffian equations with different fractional derivatives and construct the tensor representation of autonomous fractional Birkhoffian equations. Further, we study the relationship among the fractional Bikhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system and give the transformation conditions. And furthermore, as applications of the fractional Birkhoffian method, we construct five kinds of fractional dynamical models, which include the fractional Lotka biochemical oscillator model, the fractional Whittaker model, the fractional Hojman-Urrutia model, the fractional Hénon-Heiles model and the fractional Emden model.

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Symplectic Schemes for Birkhoffian System

Cited by 20 publications

References 13 publications

Energy/dissipation-preserving Birkhoffian multi-symplectic methods for Maxwell's equations with dissipation terms

Energy/dissipation-preserving Birkhoffian multi-symplectic methods for Maxwell's equations with dissipation terms

Dynamic Symmetry Breaking and Generalized Multi-symplectic Method for Non-conservative Systems

Fractional Birkhoffian mechanics

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