CFD Online Discussion Forums - nonlinear equation with non-constant (nonlinear) and direction dependent diffusion

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

in rectangular

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

where:
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),
and
$A = 1+\left(\frac{\partial y}{\partial z}\right)^2$
$C = \left(\frac{\partial y}{\partial u}\right)^2$
and
$D = 2 \left[ \frac{\partial y}{\partial z}\frac{\partial y}{\partial u}\right] \frac{\partial^2 y}{\partial z \partial u}$
and
$v = -\frac{\partial \psi}{\partial z}$ , (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*fvm::ddt(y)
- 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 ?

Thanks
ZMM