Pseudo-symplectic Runge-Kutta methods

Aubry, Anne; Chartier, Philippe

doi:10.1007/bf02510253

Cited by 49 publications

(32 citation statements)

References 9 publications

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“…The condition (1.4) forces the symplectic RungeKutta method (1.3) to be implicit. In the interest of computation efficiency, Aubry and Chartier investigated pseudo-symplectic Runge-Kutta methods, which are explicit and conserve the symplectic structure to a certain order [1]. We also note the closely related work in [2], where the error estimate for the Lie-Poisson structure is established for integration of Lie-Poisson systems using the midpoint rule.…”

Section: ∂H(pq) ∂Q Q(t) =mentioning

confidence: 99%

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Almost symplectic Runge–Kutta schemes for Hamiltonian systems

Tan

2005

Journal of Computational Physics

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Abstract.Symplectic Runge-Kutta schemes for integration of general Hamiltonian systems are implicit. In practice the implicit equations are often approximately solved based on the Contraction Mapping Principle, in which case the resulting integration scheme is no longer symplectic. In this note we prove that, under suitable conditions, the integration scheme based on an n-step successive approximation is O(δ n+2 ) away from a symplectic scheme with δ ∈ (0, 1). Therefore, this scheme is "almost" symplectic when n is large.

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Section: ∂H(pq) ∂Q Q(t) =mentioning

confidence: 99%

“…In this note, we take a different approach from [1]. Successive approximation based upon the Contraction Mapping Principle is often used to obtain an approximate solution to y i in (1.3).…”

Section: ∂H(pq) ∂Q Q(t) =mentioning

confidence: 99%

Almost symplectic Runge–Kutta schemes for Hamiltonian systems

Tan

2005

Journal of Computational Physics

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“…Since Aubry and Chartier in [1] have studied the existence and construction of explicit RK methods with standard order p and ps-order 2 p, the methods derived by these authors are natural candidates in our search of QFI-conservation methods. The paper is organized as follows: In Section 2 we derive the h-power series expansion of Q(ψ h, f (y 0 )) − Q(y 0 ) in terms of bilinear forms of the elementary differentials of f (y) at y 0 .…”

Section: Introductionmentioning

confidence: 99%

Approximate preservation of quadratic first integrals by explicit Runge–Kutta methods

et al. 2008

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The approximate preservation of quadratic first integrals (QFIs) of differential systems in the numerical integration with Runge-Kutta (RK) methods is studied. Conditions on the coefficients of the RK method to preserve all QFIs up to a given order are obtained, showing that the pseudosymplectic methods studied by Aubry and Chartier (BIT 98(3):439-461, 1998) of algebraic order p preserve QFIs with order q = 2 p. An expression of the error of conservation of QFIs by a RK method is given, and a new explicit sixstage formula with classical order four and seventh order of QFI-conservation is obtained by choosing their coefficients so that they minimize both local truncation and conservation errors. Several formulas with algebraic orders 3 and 4 and different orders of conservation have been tested with some problems with quadratic and general first integrals. It is shown that the new fourth-order explicit method preserves much better the qualitative properties of the flow than the standard fourth-order RK method at the price of two extra 256 M. Calvo et al. function evaluations per step and it is a practical and efficient alternative to the fully implicit methods required for a complete preservation of QFIs.

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“…In [23,24] explicit methods have been proposed, which preserve the symplecticity with the accuracy higher than the accuracy of the numerical solution itself. In the following section, we present the results of a comparative study of solving Equation (16) by the Runge-Kutta (RK) method of order four and by the pseudo-symplectic method PS36 [23] that has the third-order accuracy of the numerical solution and the sixth-order accuracy in preserving the simplecticity of the system. These test calculations demonstrate the effectiveness of using pseudo-symplectic integrators for solving the problem considered here.…”

Section: Propertymentioning

confidence: 99%

Numerical study of an inviscid incompressible flow through a channel of finite length

Говорухин

Ilin

2008

Numerical Methods in Fluids

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SUMMARYA two-dimensional inviscid incompressible flow in a rectilinear channel of finite length is studied numerically. Both the normal velocity and the vorticity are given at the inlet, and only the normal velocity is specified at the outlet. The flow is described in terms of the stream function and vorticity. To solve the unsteady problem numerically, we propose a version of the vortex particle method. The vorticity field is approximated using its values at a set of fluid particles. A pseudo-symplectic integrator is employed to solve the system of ordinary differential equations governing the motion of fluid particles. The stream function is computed using the Galerkin method. Unsteady flows developing from an initial perturbation in the form of an elliptical patch of vorticity are calculated for various values of the volume flux of fluid through the channel. It is shown that if the flux of fluid is large, the initial vortex patch is washed out of the channel, and when the flux is reduced, the initial perturbation evolves to a steady flow with stagnation regions.

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Pseudo-symplectic Runge-Kutta methods

Cited by 49 publications

References 9 publications

Almost symplectic Runge–Kutta schemes for Hamiltonian systems

Almost symplectic Runge–Kutta schemes for Hamiltonian systems

Approximate preservation of quadratic first integrals by explicit Runge–Kutta methods

Numerical study of an inviscid incompressible flow through a channel of finite length

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