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MathGroup Archive 2005

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Re: NDSolve and differential equation system

  • To: mathgroup at smc.vnet.net
  • Subject: [mg54027] Re: NDSolve and differential equation system
  • From: "Jens-Peer Kuska" <kuska at informatik.uni-leipzig.de>
  • Date: Tue, 8 Feb 2005 05:31:02 -0500 (EST)
  • Organization: Uni Leipzig
  • References: <cu1vjh$ltj$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,

NDSolve[] has a step size control, any numerical integration
assume that the right hand side has smooth derivatives and
use this information to estimate the step size.

Your UnitStep[] functions introduce nonsmooth derivatives
at the right hand side and mix up the step size control.

You can a) use the wonderful advanced interface of NDSolve[]
to use a constant step size or b) use smooth functions like
Erf[] or ArcTan[] instead of UnitStep[]

Regards
  Jens


"Christian Moosmann" <some@Adress> schrieb im Newsbeitrag 
news:cu1vjh$ltj$1 at smc.vnet.net...
> Hi MathGroup
>
> I have a question concerning NDSolve and a system of differential 
> equations.
>
> consider an equation system A x'[t] + (K1 + v[t] * K2).x[t]=B, where
> A,K1,K2 are Matrices, B and x are vectors and v[t] is a defined skalar
> function. I want to compute this system with NDSolve, however, it takes
> a very long time.
>
> To try it you should use Theodor Gray's ShowStatus function in the 
> frontend.
>
> ShowStatus[status_] :=
>     LinkWrite[$ParentLink,
>       SetNotebookStatusLine[FrontEnd`EvaluationNotebook[],
>         ToString[ status ] ] ];
>
> Dim = 40;
>
> I use random Matrices here. They show the same basic behaviour as the
> matrices I usually use (those are also dense), so this problem should
> not be related directly to the matrices.
>
> K1 = Table[Random[], {i, Dim}, {j, Dim}];
> K2 = Table[Random[], {i, Dim}, {j, Dim}];
> Ainv = Inverse[Table[Random[], {i, Dim}, {j, Dim}]];
> B = Table[Random[], {i, Dim}];
>
> tStart = 0.;
> tEnd = 1;
>
> At first we solve without the function v[t]:
>
> Timing[NDSolve[{D[ x[t], t] + Ainv.( K1 + K2).x[t] == B  ,
>       x[tStart] == Table[0. , {Dim}]} , x , { t, tStart, tEnd },
>     SolveDelayed -> True, StepMonitor :> ShowStatus[t] ] ]
>
> {0.080987 Second, {{x -> InterpolatingFunction[{{0., 1.}}, <>]}}}
>
> is quite fast, now we would like to include the function:
>
> v[t_] := UnitStep[t - 0.2]*4*(t - 0.2) - UnitStep[t - 0.45]*4*(t - 0.45);
>
> Timing[NDSolve[{D[ x[t], t] + Ainv.( K1 + v[t]*K2).x[t] == B  ,
>       x[tStart] == Table[0. , {Dim}]} , x , { t, tStart, tEnd },
>     SolveDelayed -> True, StepMonitor :> ShowStatus[t] ] ]
>
> {137.341 Second, {{x -> InterpolatingFunction[{{0., 1.}}, <>]}}}
>
> Well, this is a factor of > 1000, and what I would like to compute is at
> least Dimension 100. If you use the stepmonitor you also may notice,
> that the time counts up, then stops for some time, counts up again...
>
>
> Does anyone have any suggestions how to deal with that?
>
> Thanks in advance
>
> Christian
> 



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