Splitting multisymplectic integrators for Maxwell’s equations

Kong, Linghua; Hong, Jialin; Zhang, Jingjing

doi:10.1016/j.jcp.2010.02.010

Cited by 73 publications

(45 citation statements)

References 25 publications

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“…At the end of last century, MIs have been put forward and applied to large numbers of partial differential equations (PDEs), such as wave equation [2,12,15], nonlinear Schrödinger-type equations [4,6], Dirac equation [5], Maxwell's equations [9], RLW equation [7], Klein-Gordon-Schrödinger equation [19]. The most important character of MIs is its multi-symplecticity, and other conservative properties are preserved excellently despite of not exactly [1,5,8].…”

Section: Proposition 1 the Determined Problem (1) Satisfies The Follmentioning

confidence: 99%

Multi-symplectic preserving integrator for the Schrödinger equation with wave operator

Wang

Kong

Zhang³

et al. 2015

Applied Mathematical Modelling

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In this article, we discuss the conservation laws for the nonlinear Schrödinger equation with wave operator under multi-symplectic integrator (MI). First, the conservation laws of the continuous equation are presented and one of them is new. The multisymplectic structure and MI are constructed for the equation. The discrete conservation laws of the numerical method are analyzed. It is verified that the proposed MI can stably simulate the Hamiltonian PDEs excellently over long-term. It is more accurate than some energy-preserving schemes though they are of the same accuracy. Moreover, the residual of mass is less than energy-preserving schemes under the same mesh partition in a long time.

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Section: Proposition 1 the Determined Problem (1) Satisfies The Follmentioning

confidence: 99%

Multi-symplectic preserving integrator for the Schrödinger equation with wave operator

Wang

Kong

Zhang³

et al. 2015

Applied Mathematical Modelling

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“…Splitting scheme has been used for integrating partial differential equations [23,26] and has been applied to discrete Hamiltonian system [22] and multi-symplectic systems [8,18,25]. The basic idea of splitting scheme is to decompose the original problem into subproblems which are easier to solve than the original one.…”

Section: Splitting Swcm and Mswcm For The 2d-nlsementioning

confidence: 99%

Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

Zhu

Chen

Song

et al. 2011

Applied Numerical Mathematics

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“…At the computational side, the extension of symplectic integrators to partial differential equations (PDEs) led to multisymplectic integrators [44][45][46][47][48][49]. They have been used to solve a wide range of problems in physics [16,[50][51][52][53].…”

Section: Introductionmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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Splitting multisymplectic integrators for Maxwell’s equations

Cited by 73 publications

References 25 publications

Multi-symplectic preserving integrator for the Schrödinger equation with wave operator

Multi-symplectic preserving integrator for the Schrödinger equation with wave operator

Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

A review of some geometric integrators

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