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nonlinear equation with non-constant (nonlinear) and direction dependent diffusion

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Old   January 28, 2013, 12:50
Question nonlinear equation with non-constant (nonlinear) and direction dependent diffusion
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Mieszko Młody
Join Date: Mar 2009
Location: POLAND, USA
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Dear Foamers,

I am trying to solve/implement nonlinear equation with non-constant (nonlinear) and direction dependent diffusion.
It looks like that:

Equation is solved for y in rectangular (u,z) domain:
\frac{\partial y}{\partial t} + \frac{\partial  \psi}{\partial z}  =  \frac{\partial^2 y}{\partial z^2} +  \left[ \frac{1+(\frac{\partial  y}{\partial z})^2}{(\frac{\partial y}{\partial  u})^2}\right]  \frac{\partial^2 y}{\partial u^2}
 - 2 \left[ \frac{\frac{\partial y}{\partial z}}{\frac{\partial  y}{\partial u}}\right] \frac{\partial^2 y}{\partial z \partial u}

I decided to solve it as follows:

fvScalarMatrix yEqn
- U.component(1)
- fvm::laplacian(nu + (A/C)*nu2, y)
+ D/C

nu and nu2 are defined to act in z and u directions respectively
nu = (1 0 0 0 0 0 0 0 0) and nu2 = (0 0 0 0 1 0 0 0 0 0),
A = 1+\left(\frac{\partial y}{\partial z}\right)^2
C = \left(\frac{\partial y}{\partial  u}\right)^2
D = 2 \left[ \frac{\partial y}{\partial z}\frac{\partial  y}{\partial u}\right] \frac{\partial^2 y}{\partial z \partial u}
v = -\frac{\partial  \psi}{\partial z}, (vertical velocity)
where A, C and D are solved explicitly.
But solver had a problem with it, because C were getting too large.

So I decided to multiply above equation by C and get:

fvScalarMatrix yEqn
- C*U.component(1)
- fvm::laplacian(C*nu + A*nu2, y)
+ D

Here I got solution, somehow similar to the one it should converge, but not exactly.
Problem seems to be in "C*fvm::ddt(y)" term, because it gives me the same answer if I have just "fvm::ddt(y)" ...

Any idea, what can be here wrong, or how to make it better ?

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direction dependent diff, non-constant diffusion, nonlinear diffusion, nonlinear equation

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