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MathGroup Archive 2002

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Re: Checking the Results of NDSolve

  • To: mathgroup at smc.vnet.net
  • Subject: [mg37268] Re: Checking the Results of NDSolve
  • From: Tom Burton <tburton at brahea.com>
  • Date: Mon, 21 Oct 2002 02:29:36 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

On 10/18/02 2:26 AM, in article aook3e$pij$1 at smc.vnet.net, "Thomas Steger"
<steger at uni-greifswald.de> wrote:

> I use NDSolve to approximate a 6-dimensional and highly non-linear
> dynamic system numerically.
> 
> QUESTION: What are usual (and simple) techniques to check the
> reliability of the resulting numerical solution?

I would check the the boundary conditions and then plot the residual error
from the differential equation. Here is an example with all output deleted.

possibilities = NDSolve[{y[x]^4 + y'[x]^3 == 0, y[0] - 1 == 0},
                        y[x], {x, 0, 1}]
   
yy[x_] = y[x] /. possibilities

SHOW RESULTS

Needs["Graphics`Colors`"];
SetOptions[Plot, 
  PlotStyle -> ({Thickness[0.01], #1} & ) /@ {Red,Green,Blue}
 ]; 
     
Plot[Evaluate[Re[yy[x]]], {x, 0, 1}]

Plot[Evaluate[Im[yy[x]]], {x, 0, 1}]

CHECK INITIAL CONDITIONS

yy[0] - 1

PREPARE AND SHOW RESIDUAL

res[x_] = yy[x]^4 + yy'[x]^3

Plot[Evaluate[Re[res[x]]], {x, 0, 1}]

Plot[Evaluate[Im[res[x]]], {x, 0, 1}]
  
Plot[Evaluate[Re[res[x]]], {x, 0, 0.001}, PlotRange -> All]
  
Plot[Evaluate[Im[res[x]]], {x, 0, 0.001}, PlotRange -> All]



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