A symplectic integration algorithm for separable Hamiltonian functions

Candy, J.; Rozmus, W.

doi:10.1016/0021-9991(91)90299-z

Cited by 238 publications

(169 citation statements)

References 5 publications

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“…This factorization scheme has been independently derived many times in the context of symplectic integrators [16,17]. The above derivation was first published by by Creutz and Gocksch [10] in 1989.…”

Section: Fourth Order Operator Splittingsmentioning

confidence: 96%

Fourth order gradient symplectic integrator methods for solving the time-dependent Schrödinger equation

Chin

Chen

2001

The Journal of Chemical Physics

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We show that the method of splitting the operator e ǫ(T +V ) to fourth order with purely positive coefficients produces excellent algorithms for solving the time-dependent Schrödinger equation. These algorithms require knowing the potential and the gradient of the potential. One 4th order algorithm only requires four Fast Fourier Transformations per iteration. In a one dimensional scattering problem, the 4th order error coefficients of these new algorithms are roughly 500 times smaller than fourth order algorithms with negative coefficient, such as those based on the traditional Ruth-Forest symplectic integrator. These algorithms can produce converged results of conventional second or fourth order algorithms using time steps 5 to 10 times as large. Iterating these positive coefficient algorithms to 6th order also produced better converged algorithms than iterating the Ruth-Forest algorithm to 6th order or using Yoshida's 6th order algorithm A directly.

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Section: Fourth Order Operator Splittingsmentioning

confidence: 96%

Fourth order gradient symplectic integrator methods for solving the time-dependent Schrödinger equation

Chin

Chen

2001

The Journal of Chemical Physics

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“…. p should be taken from tables availables, e.g., in [5], [12]. Note that for p = 2, we have a 1 = a 2 = 1/2 and b 1 = 0, b 2 = 1, which corresponds to the leap-frog scheme.…”

Section: Consistency Criteria For Time-domain Fem Schemesmentioning

confidence: 99%

Charge-conserving FEM–PIC schemes on general grids

Pinto

Jund

Salmon

et al. 2014

Comptes Rendus. Mécanique

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In this article we aim at proposing a general mathematical formulation for charge conserving finite element Maxwell solvers coupled with particle schemes. In particular, we identify the finite element continuity equations that must be satisfied by the discrete current sources for several classes of time domain Vlasov-Maxwell simulations to preserve the Gauss law at each time step, and propose a generic algorithm for computing such consistent sources. Since our results cover a wide range of schemes (namely curl-conforming finite element methods of arbitrary degree, general meshes in 2 or 3 dimensions, several classes of time discretization schemes, particles with arbitrary shape factors and piecewise polynomial trajectories of arbitrary degree), we believe that they provide a useful roadmap in the design of high order charge conserving FEM-PIC numerical schemes.

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“…1-3. Based on a discretization of the variational principle rather than the equations of motion 9 , these algorithms exactly conserve a symplectic structure [10][11][12][13] . As a consequence of this 9,10,14,15 , they exhibit very good long time conservation properties, and numerical solutions stay close to exact dynamics, even at large time-step.…”

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confidence: 99%

Gauge Properties Of The Guiding Center Variational Symplectic Integrator

Squire¹

2012

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Variational symplectic algorithms have recently been developed for carrying out long-time simulation of charged particles in magnetic fields [1][2][3] . As a direct consequence of their derivation from a discrete variational principle, these algorithms have very good long-time energy conservation, as well as exactly preserving discrete momenta. We present stability results for these algorithms, focusing on understanding how explicit variational integrators can be designed for this type of system. It is found that for explicit algorithms an instability arises because the discrete symplectic structure does not become the continuous structure in the t → 0 limit. We examine how a generalized gauge transformation can be used to put the Lagrangian in the "antisymmetric discretization gauge," in which the discrete symplectic structure has the correct form, thus eliminating the numerical instability. Finally, it is noted that the variational guiding center algorithms are not electromagnetically gauge invariant. By designing a model discrete Lagrangian, we show that the algorithms are approximately gauge invariant as long as A and φ are relatively smooth. A gauge invariant discrete Lagrangian is very important in a variational particle-in-cell algorithm where it ensures current continuity and preservation of Gauss's law 4 .

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A symplectic integration algorithm for separable Hamiltonian functions

Cited by 238 publications

References 5 publications

Fourth order gradient symplectic integrator methods for solving the time-dependent Schrödinger equation

Fourth order gradient symplectic integrator methods for solving the time-dependent Schrödinger equation

Charge-conserving FEM–PIC schemes on general grids

Gauge Properties Of The Guiding Center Variational Symplectic Integrator

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