Runge-kutta schemes for Hamiltonian systems

Sanz‐Serna, J. M.

doi:10.1007/bf01954907

Cited by 402 publications

(246 citation statements)

References 10 publications

(14 reference statements)

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“…A closely related result is that if M = 0, the method preserves symplectic behaviour for Hamiltonian problems [13].…”

Section: Non-linear Stabilitymentioning

confidence: 96%

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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“…A closely related result is that if M = 0, the method preserves symplectic behaviour for Hamiltonian problems [13].…”

Section: Non-linear Stabilitymentioning

confidence: 96%

Forty‐five years of A‐stability

Butcher¹

2008

AIP Conference Proceedings

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“…This method requires special time integrators called symplectic integrators [19]. This approach is known as the Lagrangian flow field specification, where one follows a single trajectory in phase space.…”

Section: Lagrangian Versus Eulerian Picturementioning

confidence: 99%

A Novel Scheme for Liouville’s Equation with a Discontinuous Hamiltonian and Applications to Geometrical Optics

et al. 2016

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A novel scheme is developed that computes numerical solutions of Liouville's equation with a discontinuous Hamiltonian. It is assumed that the underlying Hamiltonian system has well-defined behaviour even when the Hamiltonian is discontinuous. In the case of geometrical optics such a discontinuity yields the familiar Snell's law or the law of specular reflection. Solutions to Liouville's equation should be constant along curves defined by the Hamiltonian system when the right-hand side is zero, i.e., no absorption or collisions. This consideration allows us to derive a new jump condition, enabling us to construct a first-order accurate scheme. Essentially, the correct physics is built into the solver. The scheme is tested in a two-dimensional optical setting with two test cases, the first using a single jump in the refractive index and the second a compound parabolic concentrator. For these two situations, the scheme outperforms the more conventional method of Monte Carlo ray tracing.

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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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Runge-kutta schemes for Hamiltonian systems

Cited by 402 publications

References 10 publications

Forty‐five years of A‐stability

Forty‐five years of A‐stability

A Novel Scheme for Liouville’s Equation with a Discontinuous Hamiltonian and Applications to Geometrical Optics

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

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