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Re: explicit 2nd order scheme


OK Bruno,

thank' s for this explaination.

Just two points:
i) this "explicit 2nd order finite difference scheme" is called half-step
"leap frog" scheme
(Hocknet 1970, Potter 1972, cited in
*Computer Simulation Of Liquids** by Allen
<http://www.angusrobertson.com.au/by/allen/549/>, Tildesley
<http://www.angusrobertson.com.au/by/tildesley/116428/>, M. P.
Allen<http://www.angusrobertson.com.au/by/m-p-allen/517561/>and D.
J. Tildesley <http://www.angusrobertson.com.au/by/d-j-tildesley/587821/>*)
ii) Because the scheme is a "leap frog"  scheme, the velocity is not
computed at the same time as
the accelerations and the positions...
Accelerations and positions are known at time *t
*and velocities are known at the time *t-dt/2* (or *t+dt/2*)
In YADE, only the "sign"  of the velocity is used (for damping).
Nevertheless, in YADE, we use velocities  at time *t-dt/2 *to * *"damp"
computed at time *t*.
The error made here is probably very very small.....


2009/9/17 Bruno Chareyre <bruno.chareyre@xxxxxxxxxxx>

> Re-sent to users list. Sorry for doublets.
> Jerome Duriez a écrit :
>> And how are velocities (in both schemes) computed ?
>> If they are computed by the same mean, I can still not understand at all
>> what is the difference between these two shemes, except the fact that "2nd
>> order" sounds more serious than "1st order"...
>>  Héhé...
> In first order scheme, you would use current velocity for p(t+dt) = p(t) +
> v(t)*dt, THEN you would compute v(t+dt)=..., in 2nd order, you use the
> updated (second) value instead. A while ago, Yade was using the 1st order
> scheme for rotations, and it was the reason why we needed so small time
> steps.
> It seems many people have problems with this simple 2nd order explicit
> scheme. I send the equations in the attached file, with time step
> determination. I hope (should I?) it will stop endless discussions in my
> office about this. ;)
> You will find the exact same scheme in any Cundall's paper, even if he
> doesn't explain it exactly the same way perhaps.
> You will not find the "leap-frog" naming in Cundall's papers though, and
> personnaly, I don't know what leap-frog is. I only know "explicit 2nd order
> finite difference scheme", and this is what's in Yade.
> Bruno
> --
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> Chareyre Bruno
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