[ODE] Other integrators

[Kenneth Holmlund]

Russ Smith wrote:

>>higher order schemes genererally dont use results computed in
>>previous iterations, at least not for these kinds of highly
>>impulsive systems. you would have to compute multiple 'acceleration'
>>values in a single time step. the entire simulation runs N times
>>slower because of this.
>>
Yes, but on the other hand it might not even be particularly important 
to use
higher order integration when the system changes state frequently (e.g. 
collisions).
In my previous life I was doing Langevin simulations and for the system 
I was
studying (Brownian, collission dominated dynamics) you can actually 
prove that
the term in front´of the dt^2 is zero, so the first order approximation 
is just fine.
I have no idea how a formal analysis of the LCP system would look, but I
would suspect that the first order approximation is a fairly good one.

I haven't tested it but I'm quite sure ODE (as well as most other 
toolkits) has
pretty lousy performance when e.g. simulating a bound state like a satellite
orbiting earth. In these cases one can often use some specific semi-analytic
method in combination with ODE, but it would probably also work just fine
if you use ODE with higher order integration based on saved 
accelerations and
velocities from earlier time steps. However, you'd have to be careful not
to integrate between states, whenever there's an impulse change, since that
would completely ruin your state change behaviour.

>actually i'm not sure what a higher order time stepping scheme
>will look like in this case. i'd have to sit down and read some
>papers and write some equations. it's not a problem i have
>considered yet.
>
One thing is how it will look like, but the important thing is to 
estimate whether
it is at all worth going into higher order here. Using an adaptive time 
step for
colissions might as well do the job (though the real-time constraint 
sets a limit
to that). Do you know of any good references in this particular area?

/Kenneth

-- 
Kenneth Holmlund
Director VRlab
HPC2N, Umeå University, Sweden
holmlund@hpc2n.umu.se
T: +46-90-786 9655
C: +46-70-631 5520