A note on pseudo-symplectic Runge-Kutta methods

Aubry, Anne; Chartier, Philippe

doi:10.1007/bf02510415

Cited by 5 publications

(4 citation statements)

References 1 publication

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“…h is a symplectic integrator, then ψ E,h is a pseudo-symplectic method [1]. We will denote this class of methods as ψ [2(n+m)] E,h (2n; s; r): an integrator of order 2(n + m) obtained by extrapolation from a basic scheme of order 2n, evaluated s times, and which preserves the geometric character of the solution up to order r. In fact, it is possible to build methods of order 2(n + m) which preserve geometric properties up to order 4(n + m) + 1 simply by canceling r 4n+2j for j = 0, 1, .…”

Section: Extrapolation Of Geometric Integratorsmentioning

confidence: 99%

Raising the order of geometric numerical integrators by composition and extrapolation

Blanes

Casas

2005

Numer Algor

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We analyse composition and polynomial extrapolation as procedures to raise the order of a geometric integrator for solving numerically differential equations. Methods up to order sixteen are constructed starting with basic symmetric schemes of order six and eight. If these are geometric integrators, then the new methods obtained by extrapolation preserve the geometric properties up to a higher order than the order of the method itself. We show that, for a number of problems, this is a very efficient procedure to obtain high accuracy. The relative performance of the different algorithms is examined on several numerical experiments.

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Section: Extrapolation Of Geometric Integratorsmentioning

confidence: 99%

Raising the order of geometric numerical integrators by composition and extrapolation

Blanes

Casas

2005

Numer Algor

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“…The resultant piecewise continuous polynomial approximation of vorticity is employed to derive Galerkin's system of equations for ψ ij (t). For integration (9.1) we use the pseudo-symplectic integrator by Aubry & Chartier (1998). When a particle leaves D, a new particle with the vorticity equal to its boundary value (2.5) is introduced at ∂D in (with the same y); thus the total number of particles remains constant.…”

Section: Numerical Proceduresmentioning

confidence: 99%

Planar inviscid flows in a channel of finite length: washout, trapping and self-oscillations of vorticity

2010

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The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.

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“…However, these symplectic integrators are implicit and, in practice, cannot be applied to the problem considered here. 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.…”

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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A note on pseudo-symplectic Runge-Kutta methods

Cited by 5 publications

References 1 publication

Raising the order of geometric numerical integrators by composition and extrapolation

Raising the order of geometric numerical integrators by composition and extrapolation

Planar inviscid flows in a channel of finite length: washout, trapping and self-oscillations of vorticity

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

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