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I am looking for a way of getting some intuition for some Convection Diffusion Equation. I would like to play around with it, simulate it, change its IC/BCs, having some interactive graphs and stuff.

Do you know of any free software that does the job? Thank you!

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    @macydanim - But I am still interested :-) Thank you2013-01-21

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Dynare on top of Octave should do the Trick.

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OpenFOAM might also be a tool. But I am not sure, if this might be too difficult to begin with.

You can find it on their homepage. A quick google search offered this tutorial, see slide 17ff, where the convection-diffusion equation is solved. But I personally would start with the OpenFOAM tutorial.