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differentiation in one direction

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Old   August 1, 2011, 18:39
Default [solved]differentiation in one direction
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Mieszko Młody
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Dear Foamers,

Does anyone of you know what is the best way to dicsretize following problem:
we have: f(x,y) -- scalar function

we need: f_xxxx
(fourth order) differentiation but only in one direction.

i.e.
laplacian(f) = f_xx + f_yy
but I need only f_xx (in fact I need f_xxxx)

Thanks
ZM

Last edited by ziemowitzima; August 4, 2011 at 14:16. Reason: solved
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Old   August 2, 2011, 01:55
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Fabian
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You should use a difference quoutient, like you have propably seen for f_x. You can perform the difference quotient "four" times or you can have a look here (they have already done it for you):

http://en.wikipedia.org/wiki/Difference_quotient

Probably it should be the best for you to read the whole article, but you can also jump ahead to:

http://en.wikipedia.org/wiki/Differe....C5.83th_Order
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Old   August 2, 2011, 11:10
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Thanks
This I know.
I didn't put it explicit in my question but I need to know how to make it in OpenFOAM.

-Best
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Old   August 2, 2011, 17:52
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so far I found such explicit solution:

volVectorField gradT=fvc::grad(T); // (T_x, T_y)
volScalarField Txxxx = gradT.component(0); // T_x
gradT = fvc::grad(Txxxx); // (T_xx, T_xy)
Txxxx = gradT.component(0); // T_xx
gradT = fvc::grad(Txxxx); // (T_xxx, T_xxy)
Txxxx = gradT.component(0); // T_xxx
gradT = fvc::grad(Txxxx); // (T_xxxx, T_xxxy)
Txxxx = gradT.component(0); // T_xxxx

Maybe someone have some better idea ?
Is it possible to descritize T_xxxx implicitly ?

-ZM
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Old   August 4, 2011, 12:24
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It seems that problem can be solved (after Akidess suggestion ):

"... it sounds like you want anisotropic diffusion - why not just pass a diffusion coefficient tensor to laplacian instead of a scalar?..."
from:
Modifying the laplacian operator

So,
x - direction diffusion:
DTe1D DTe1D [ 0 2 -1 0 0 0 0 ] (3e-02 0 0 0 0 0 0 0 0 );

x,y and - directions diffusion:
DTe3D DTe3D [ 0 2 -1 0 0 0 0 ] (3e-02 0 0 0 3e-02 0 0 0 3e-02 );

for f_xxxx it can be done as follows :
suppose we have equation f_xxxx = g(x,y),
substitution: h = f_xx
so we have two "one-directional" Poison equations to solve:
h_xx = g(x,y)
f_xx = h(x,y)

in OF:
fvm::laplacian(DTe1D, h) == g
fvm::laplacian(DTe1D, f) == h
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