Hello all,

I'm trying to find a nice and neat way to numerically solve the
convective diffusion equation

da/dt = D (d^2/dx^2 + d^2/dy^2) a - v da/dx

where a is the concentration of my solute, D is the diffusion
constant, and v is the surrounding fluid velocity in the x direction.
I thought that there was a small chance that maybe someone else here
has attempted something similar.
Is it even possible to solve this equation? As always, any suggestions
would be much appreciated.

Cheers,
Dan

dantimatter wrote:
> Hello all,
> 
> I'm trying to find a nice and neat way to numerically solve the
> convective diffusion equation
> 
> da/dt = D (d^2/dx^2 + d^2/dy^2) a - v da/dx
> 
> where a is the concentration of my solute, D is the diffusion
> constant, and v is the surrounding fluid velocity in the x direction.
> I thought that there was a small chance that maybe someone else here
> has attempted something similar.
> Is it even possible to solve this equation? As always, any suggestions
> would be much appreciated.
> 
> Cheers,
> Dan

Oliver Rübenkönig's "Finite Elements for Convection-Diffusion equations" 
might be worth reading:

http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/imsTOC/Application%20Examples/Finite%20Element%20Method/AnemometerAnalysisDocu.html

Also, you should have a look at the  _IMTEK Mathematica Supplement_ (IMS)

http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/

HTH,
Jean-Marc

Hi,

With[{\[ScriptCapitalD] = 1/8, v = 1/4},
  sol = NDSolve[{
     D[u[x, y, t],
       t] == \[ScriptCapitalD] (D[u[x, y, t], {x, 2}] +
          D[u[x, y, t], {y, 2}]) - v*D[u[x, y, t], x],
     u[-1, y, t] == 0, u[1, y, t] == 0,
     u[x, -1, t] == 0, u[x, 1, t] == 0,
     u[x, y, 0] == Piecewise[{{1, Sqrt[x^2 + y^2] <= 0.5}}, 0]},
    u[x, y, t], {x, -1, 1}, {y, -1, 1}, {t, 0, 2}
    ]
  ]

may do that .. Ignore the warnings you get with the
statement above, the reason are the initial conditions
and you may have other.

Regards
   Jens

dantimatter wrote:
> Hello all,
> 
> I'm trying to find a nice and neat way to numerically solve the
> convective diffusion equation
> 
> da/dt = D (d^2/dx^2 + d^2/dy^2) a - v da/dx
> 
> where a is the concentration of my solute, D is the diffusion
> constant, and v is the surrounding fluid velocity in the x direction.
> I thought that there was a small chance that maybe someone else here
> has attempted something similar.
> Is it even possible to solve this equation? As always, any suggestions
> would be much appreciated.
> 
> Cheers,
> Dan
> 
> 

> With[{\[ScriptCapitalD] = 1/8, v = 1/4},
>   sol = NDSolve[{
>      D[u[x, y, t],
>        t] == \[ScriptCapitalD] (D[u[x, y, t], {x, 2}] +
>           D[u[x, y, t], {y, 2}]) - v*D[u[x, y, t], x],
>      u[-1, y, t] == 0, u[1, y, t] == 0,
>      u[x, -1, t] == 0, u[x, 1, t] == 0,
>      u[x, y, 0] == Piecewise[{{1, Sqrt[x^2 + y^2] <= 0.5}}, 0]},
>     u[x, y, t], {x, -1, 1}, {y, -1, 1}, {t, 0, 2}
>     ]
>   ]
>
> may do that .. Ignore the warnings you get with the
> statement above, the reason are the initial conditions
> and you may have other.

Hi Jens,

I wasn't able to plot the result.  Have you tried this and
successfully plotted it?

Cheers,
Dan

> With[{\[ScriptCapitalD] = 1/8, v = 1/4},
>   sol = NDSolve[{
>      D[u[x, y, t],
>        t] == \[ScriptCapitalD] (D[u[x, y, t], {x, 2}] +
>           D[u[x, y, t], {y, 2}]) - v*D[u[x, y, t], x],
>      u[-1, y, t] == 0, u[1, y, t] == 0,
>      u[x, -1, t] == 0, u[x, 1, t] == 0,
>      u[x, y, 0] == Piecewise[{{1, Sqrt[x^2 + y^2] <= 0.5}}, 0]},
>     u[x, y, t], {x, -1, 1}, {y, -1, 1}, {t, 0, 2}
>     ]
>   ]

ok, so i got the above to work when i ran it on a more powerful
machine. thanks for the advice.  the problem i'm having now is in
defining the boundary conditions.  what i'd like is to have a circular
'source' at which the concentration is always constant, but i don't
know about any of the other boundaries.  i guess i could say that the
concentration is zero at infinity.  any advice on how to implement
these boundary conditions??

many thanks,
dan

Sorry for interfering,
but I had a similar question before.

Using the procedure of Jens in the form

s = First[With[{\[ScriptCapitalD] = 1/8, v = 1/4},
sol = NDSolve[{D[u[x, y, t], t] ==
\[ScriptCapitalD]*(D[u[x, y, t], {x, 2}] + D[u[x, y, t],
{y, 2}]) - v*D[u[x, y, t], x],
u[-1, y, t] == 0, u[1, y, t] == 0,
u[x, -1, t] == 0, u[x, 1, t] == 0,
u[x, y, 0] == Piecewise[
{{1, Sqrt[x^2 + y^2] <= 0.5}}, 0]},
u[x, y, t], {x, -1, 1}, {y, -1, 1}, {t, 0, 2}]]]

{u[x, y, t] -> InterpolatingFunction[][x, y, t]}

extractiing the solution into a function h

h[x_, y_, t_] = u[x, y, t] /. s

InterpolatingFunction[][x, y, t]

and finally plotting the result using

t = 1;
Plot3D[h[x, y, t], {x, -1, 1}, {y, -1, 1}]

or the sequence

For[t = 1, t <= 2, t = t + 0.1, Plot3D[h[x, y, t],
{x, -1, 1}, {y, -1, 1}, PlotRange -> {0, 0.45}]]

looks very nice.

Best regards,
Wolfgang


"dantimatter" <dantimatter@gmail.com> schrieb im Newsbeitrag 
news:f943hn$m56$1@smc.vnet.net...
>
>> With[{\[ScriptCapitalD] = 1/8, v = 1/4},
>>   sol = NDSolve[{
>>      D[u[x, y, t],
>>        t] == \[ScriptCapitalD] (D[u[x, y, t], {x, 2}] +
>>           D[u[x, y, t], {y, 2}]) - v*D[u[x, y, t], x],
>>      u[-1, y, t] == 0, u[1, y, t] == 0,
>>      u[x, -1, t] == 0, u[x, 1, t] == 0,
>>      u[x, y, 0] == Piecewise[{{1, Sqrt[x^2 + y^2] <= 0.5}}, 0]},
>>     u[x, y, t], {x, -1, 1}, {y, -1, 1}, {t, 0, 2}
>>     ]
>>   ]
>>
>> may do that .. Ignore the warnings you get with the
>> statement above, the reason are the initial conditions
>> and you may have other.
>
> Hi Jens,
>
> I wasn't able to plot the result.  Have you tried this and
> successfully plotted it?
>
> Cheers,
> Dan
>
>