Stability of Runge-Kutta Methods for Trajectory Problems

Cooper, G. J.

doi:10.1093/imanum/7.1.1

Cited by 174 publications

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“…It turns out that these are exactly the same conditions which make a PRK method a symplectic integrator when applied to Hamiltonian differential equations (see for an overview on symplectic methods). This corresponds to a similar result, given by Cooper [3], for nonpartitioned Runge-Kutta methods. The conservation of angular momentum by explicit symplectic RungeKutta-Nystrom methods has been discussed before by Zhang and Skeel [18].…”

Section: F(qp)=v P H(qp) and G(qp)=-v Q H(qp)supporting

confidence: 88%

Momentum conserving symplectic integrators

Reich¹

1994

Physica D: Nonlinear Phenomena

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Sebastian Reich Momentum conserving symplectic integrators U n i v e r s i t ä t P o t s d a m AbstractIn this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).

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Section: F(qp)=v P H(qp) and G(qp)=-v Q H(qp)supporting

confidence: 88%

Momentum conserving symplectic integrators

Reich¹

1994

Physica D: Nonlinear Phenomena

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“…Although A-stability does not imply B-stability, the closely related concepts of AN and BN-stability are equivalent. The matrix M , which plays a central role in B-stability, has since become of crucial importance in the study of canonical Runge-Kutta methods [4] which respect quadratic invariants. Suppose y, Qy is a quadratic invariant, so that f (y), Qy = 0, then, noting that (6) still holds if the inner product ·, · is replaced by ·, Q· , we see that, if M = 0, y n , Qy n = y n−1 , Qy n−1 , indicating that invariance is preserved by numerical approximations.…”

Section: Non-linear Stabilitymentioning

confidence: 99%

Forty‐five years of A‐stability

Butcher¹

2008

AIP Conference Proceedings

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Abstract:We discuss two events with profound implications on the way initial value problems are solved numerically. The first was the identification of stiffness as a widely spread phenomenon affecting the ability to obtain useful results. The second was the definition of A-stability as an important approach to overcoming the effect of stiffness. Not only was the idea associated with A-stability significant in its own time but it has had long term effects including new theoretical questions as well as the tools for solving them.

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“…Quadratic invariants appear very often in applications, the conservation law of angular momentum in N body systems, the conservation of total angular momentum and kinetic energy of the rigid body motion etc. are the examples, we refer the readers interested in it to [4,7,14,16] and references therein for their more applications.…”

Section: ∇I(z)mentioning

confidence: 99%

Quadratic invariants and multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

Sun

2007

Numer. Math.

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In this paper, we study the preservation of quadratic conservation laws of Runge-Kutta methods and partitioned Runge-Kutta methods for Hamiltonian PDEs and establish the relation between multi-symplecticity of Runge-Kutta method and its quadratic conservation laws. For Schrödinger equations and Dirac equations, the relation implies that multi-sympletic RungeKutta methods applied to equations with appropriate boundary conditions can preserve the global norm conservation and the global charge conservation respectively.

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Stability of Runge-Kutta Methods for Trajectory Problems

Cited by 174 publications

References 0 publications

Momentum conserving symplectic integrators

Momentum conserving symplectic integrators

Forty‐five years of A‐stability

Quadratic invariants and multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

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