Talk:Leapfrog integration

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Use in electronic engineering

This term can also be found in the field of electronic filter realisation. A standard passive filter can be actively simulated (called functional simulation) by interchanging current and voltage in a "leapfrog" way in the equations describing the circuit. —Preceding unsigned comment added by Aphexer (talk • contribs) 18:05, 15 January 2009 (UTC)[reply]

definition of v

Should it be stated explicitly that v = dx/dt? —Preceding unsigned comment added by 128.100.76.206 (talk) 19:40, 30 July 2009 (UTC)[reply]

conservation of H

The article states that the leapfrog method "conserves a (slightly modified) energy of dynamical systems". This is a widely spread misconception and is simply not true, except in very special cases - most notably linear systems (harm. osc.). The symplecticity implies phase-space area preservation, yes - but not necessarily Hamiltonian conservation. Instead, typical systems even in one dim are non-integrable, and display more or less chaos. — Preceding comment added by 130.235.189.247 (talk • contribs) 20:14, 7 February 2013

Please propose a better formulation. If I read Hairer right, it should at least be true that a modified Hamiltonian is preserved to the order

O(e^{Lt}*\Delta t^{4})

(time symmetry kills odd powers of

\Delta t

), and that the difference between original and modified Hamiltonian is

O(\Delta t^{2})

. KAM style chaos is perfectly compatible with Hamiltionian preservation, no energy is being destroyed or created.--LutzL (talk) 10:53, 10 March 2014 (UTC)[reply]

The existence of a conserved "shadow Hamiltonian" which converges to the exact Hamiltonian in the limit of vanishing time step is quite general. A rather straightforward proof is presented in section 13 of chapter 3 of Tuckerman's Statistical Mechanics textbook, which I have now referenced in the article.^[1] A system exhibiting chaotic dynamics does not mean that there is non-conservation. Most physical dynamical systems beyond of few particles exhibit chaotic dynamics, and this is precisely why symplectic integrators are used in dynamical simulations; they help to minimize energy drift by sticking to the shadow manifold.

References

^ Tuckerman, Mark E. (2010). Statistical Mechanics: Theory and Molecular Simulation (1 ed.). Oxford University Press. pp. 121–124. ISBN 9780198525264.

Contrast with Euler-Cromer

The algorithm described in the article,

${\begin{aligned}x_{i}&=x_{i-1}+v_{i-1/2}\,\Delta t,\\[0.4em]a_{i}&=F(x_{i})\\[0.4em]v_{i+1/2}&=v_{i-1/2}+a_{i}\,\Delta t,\end{aligned}}$

or equivalently

${\begin{aligned}a_{i}&=F(x_{i})\\[0.4em]v_{i+1/2}&=v_{i-1/2}+a_{i}\,\Delta t,\\x_{i+1}&=x_{i}+v_{i+1/2}\,\Delta t,\\[0.4em]\end{aligned}}$

looks very similar to the Euler-Cromer method. In fact, they seem to be identical if F is not a function of v - in that case they only differ in the labeling of the nuisance variable v. Yet Leapfrog is a second-order method, while Euler-Cromer is first order. So something is strange here. I'm sure other readers than I may be confused, so a discussion of this might be nice to have in the article. Amaurea (talk) 12:50, 23 June 2014 (UTC)[reply]

Read the very good paper by Hairer et al. on the history and multitude of names of the Newton-Verlet-Stoermer-... method. Also on how to interpret the methods in terms of operator/vector field splittings. It seems that you are right, the time-shift of the velocity is the only difference, and it leads to a global O(h) error in the velocity. The position in Euler-Cromer should still be O(h²) for situations that can be formulated as

{\ddot {x}}(t)=\nabla P(x(t))

.--LutzL (talk) 14:57, 23 June 2014 (UTC)[reply]

Superior to RK

Using RK methods for n-body gravitational simulators is not very useful as they do not conserve energy, so after a few orbits all bodies begin to spiral either inwards or outwards when N is larger than 3. Leapfrog conserves energy better, so provides much better long term simulations. — Preceding unsigned comment added by 131.227.74.101 (talk) 12:24, 19 February 2019 (UTC)[reply]

Kick-Drift-Kick method

This method gives me different results to the midstep method even with constant timestep. Flaws? — Preceding unsigned comment added by 131.227.74.101 (talk) 13:19, 19 February 2019 (UTC)[reply]

Broken reference

This reference is broken: http://www.artcompsci.org/vol_1/v1_web/node34.html — Preceding unsigned comment added by 2A02:120B:2C67:1960:1C5D:82C3:B66A:106F (talk) 08:32, 13 January 2022 (UTC)[reply]

[1] Tuckerman, Mark E. (2010). Statistical Mechanics: Theory and Molecular Simulation (1 ed.). Oxford University Press. pp. 121–124. ISBN 9780198525264.

[1]