Structure-Preserving Numerical Methods for Infinite-Dimensional Birkhoffian Systems

Su, Hongling; Li, Shengtai

doi:10.1007/s10915-014-9958-2

Cited by 6 publications

(4 citation statements)

References 18 publications

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“…孔新雷等人 [17,36] 构造了Birkhoff系统的离散变分差分格式, 分别应用于单自由度阻尼振动系统与最优控制问题, 提出了Birkhoff系统的离散最优控制理论. 苏红玲等人 [37] 采用多辛Birkhoff系统与保辛算法, 对弦振动问题进行了数值求解; 建立了无限维Birkhoff系统的生成泛函法 [38] , 成功应用于声波方程求解与耗散电磁场 :…”

Section: 率模型的随机系统数值分析方法unclassified

Random and interval uncertain symplectic methods for linear Birkhoff equations and their comparison study

Zhu¹,

Qiu²

2020

Sci. Sin.-Phys. Mech. Astron.

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Section: 率模型的随机系统数值分析方法unclassified

Random and interval uncertain symplectic methods for linear Birkhoff equations and their comparison study

Zhu¹,

Qiu²

2020

Sci. Sin.-Phys. Mech. Astron.

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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 non-autonomous Birkhoffian system (2) with the infinite dimensional Birkhoffian formulation (42) preserves the symplectic conservation law (44). According to the generating functional theory developed in our work [25], it implies that there exists a functional…”

Section: A Novel Birkhoffian Scheme Applied To Non-autonomous Birkhofmentioning

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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Symplectic central difference scheme for quasi-linear autonomous Birkhoffian systems

Qiu

2023

Applied Mathematics and Computation

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Structure-Preserving Numerical Methods for Infinite-Dimensional Birkhoffian Systems

Cited by 6 publications

References 18 publications

Random and interval uncertain symplectic methods for linear Birkhoff equations and their comparison study

Random and interval uncertain symplectic methods for linear Birkhoff equations and their comparison study

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

Symplectic central difference scheme for quasi-linear autonomous Birkhoffian systems

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