[gmx-users] integration scheme

[Erik Lindahl]

The Nose-Hoover extended ensemble equation of motion relies on

A = F/M - Xi*V          [1]

When combined with the leapfrog integrator,

V(t+1/2) = V(t-1/2) + dt*A(t)             [2]
X(t+1) = X(t) + dt*V(t+1/2)                [3]

the acceleration should be evaluated at time=t. This is trivial for the  
forces, and for the velocities you can simply use the fact that

V(t)=1/2*[V(t-1/2)+V(t+1/2)]

(this is not even an approximation for verlet-class integrators) Insert  
this in equation [1] (including the calculation of temperature used to  
get Xi), and then use this expression to solve equation [2] for  
V[t+1/2] . That gives you leap-frog with extended ensembles :-)

Now, that's the theory - in practice you run into problems due to  
features like acceleration groups, and all nodes don't have all  
velocities at the point of integration, so by using the temperature of  
last step we avoid doing extra communication.

Cheers,

Erik

On Sunday, May 11, 2003, at 17:36 America/Los_Angeles, Lianqing Zheng  
wrote:

> Thanks, Erik! Do you mean that you use a(r(t),v(t-0.5*dt)) instead of
> a(r(t),v(t)) in the current version of Gromacs? I still don't get it  
> how
> it was in the first version. Could you please use equations to express  
> it?
>
> Sorry for the slow understanding... :D Thanks!
>
> Lianqing
>
>
> On Sun, 11 May 2003, Erik Lindahl wrote:
>
>> Hi,
>>
>> Well, actually you can - the integration just becomes an equation that
>> is straightforward to solve :-)
>>
>> Our first version of Nose-Hoover was implemented that way, but we  
>> since
>> changed it to use the temperature calculated from velocities last  
>> step.
>> The reason for this is that it would otherwise be necessary to do  
>> extra
>> communication when running in parallel, and the error is extremely
>> small.
>>
>> I might add a switch so you turn this optimization off, but it will
>> only make a difference if you are integrating something like a single
>> harmonic oscillator...
>>
>> Cheers,
>>
>> Erik
>>
>> On Friday, May 9, 2003, at 12:22 America/Los_Angeles, Lianqing Zheng
>> wrote:
>>
>>> Dear Gromacs pals:
>>>
>>> This may be trivial, but I am curious. When Nose-Hoover temperature
>>> coupling is used, the accelerations are dependent on velocities, then
>>> regular leap-frog algorithm can't solve this kind of equations. This  
>>> is
>>> because:
>>>
>>> v(t+0.5*dt) = v(t-0.5*dt) + a(t)*dt
>>>
>>> however a(t) depends on v(t), which is unknown at this time.
>>>
>>> How does Gromacs do with it?
>>>
>>> Thanks!
>>>
>>> Lianqing
>>>
>>>
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>> ---------------------------------------------------------------------- 
>> --
>> -----
>> Erik Lindahl, MSc, PhD     <lindahl at stanford.edu>
>> D109, Fairchild Building
>> Dept. Structural Biology, Stanford University School of Medicine
>> Tel. 650-7250754    Fax. 650-7238464
>>
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------------------------------------------------------------------------ 
-----
Erik Lindahl, MSc, PhD     <lindahl at stanford.edu>
D109, Fairchild Building
Dept. Structural Biology, Stanford University School of Medicine
Tel. 650-7250754    Fax. 650-7238464