Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods

Blanes, Sergio; Moan, P. C.

doi:10.1016/s0377-0427(01)00492-7

Cited by 222 publications

(310 citation statements)

References 24 publications

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“…However, recently symplectic partitioned Runge-Kutta schemes have been constructed for the nonlinear Schrödinger equation which maybe could outperform the splitstep4 scheme used here, as reported in [9]. In further studies, these new split step schemes should also be compared to exponential integrators.…”

Section: Fourth Order Split Step Schemementioning

confidence: 92%

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

Berland

Islas

Schober

2007

Journal of Computational Physics

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The cubic nonlinear Schrödinger (nls) equation with periodic boundary conditions is solvable using Inverse Spectral Theory. The "nonlinear" spectrum of the associated Lax pair reveals topological properties of the nls phase space that are difficult to assess by other means. In this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach. The initial condition used is a perturbation of the unstable plane wave solution, which is difficult to numerically resolve. Our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step, while a lower order multisymplectic is not as accurate and too slow to compete.

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Section: Fourth Order Split Step Schemementioning

confidence: 92%

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

Berland

Islas

Schober

2007

Journal of Computational Physics

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“…In Figure 1, we compare the efficiency of the Chin scheme to two other splitting methods: the Yoshida scheme [10,30] and the optimised fourth order Runge-KuttaNyström scheme (RKN4) of Blanes and Moan [4]. We also compare with a fourth order commutator-free scheme [27], using the Arnoldi process to approximate the matrix exponentials.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…One example is Runge-Kutta-Nyström methods [4,12], which apply when commutators of the operators satisfy [B, [B, [A, B]]] = 0. This reduces the number of order conditions, and gives more freedom to the choice of method coefficients.…”

Section: Introductionmentioning

confidence: 99%

Stiff convergence of force-gradient operator splitting methods

Kieri

2015

Applied Numerical Mathematics

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Stiff convergence of force-gradient operator splitting methods. Applied Numerical Mathematics AbstractWe consider force-gradient, also called modified potential, operator splitting methods for problems with unbounded operators. We prove that force-gradient operator splitting schemes retain their classical orders of accuracy for linear time-dependent partial differential equations of parabolic and Schrödinger types, provided that the solution is sufficiently regular.

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“…All of these studies rely on the structure of the Lie algebra generated by kinetic and potential energy. Bases for this Lie algebra have been constructed, more or less by hand, for small orders [5,6,20]. In particular, Murua [20] associates a unique tree of a certain type to each independent order condition of symplectic Runge-Kutta-Nyström methods (very closely related to the problem considered here), and enumerates these up to order 6.…”

Section: Introduction Classes Of Lie Algebrasmentioning

confidence: 99%

The algebraic entropy of classical mechanics

McLachlan

Ryland

2003

Journal of Mathematical Physics

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We describe the 'Lie algebra of classical mechanics', modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. It is a polynomially graded Lie algebra, a class we introduce. We describe these Lie algebras, give an algorithm to calculate the dimensions c n of the homogeneous subspaces of the Lie algebra of classical mechanics, and determine the value of its entropy lim n→∞ c

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Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods

Cited by 222 publications

References 24 publications

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

Stiff convergence of force-gradient operator splitting methods

The algebraic entropy of classical mechanics

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