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Hello,
The following PDE is defined in a non-square 2D (x-y) cartesian grid (20 x 10)
domain
$$ \frac{\partial}{\partial t} (y^2 \phi(x,y,t)) = D \nabla^2 \phi(x,y,t) $$
wherein $y^2$ is the 'y' co-ordinate of my 2D domain.
The FiPy representation of an analogous type of equation is as follows:
$$ \frac{\partial}{\partial t} (\rho \phi(x,y,t)) = D \nabla^2 \phi(x,y,t) $$
TransientTerm(coeff=rho) == DiffusionTerm(coeff=D)
By matching coefficients, we can see that my $\rho = y^2$
I am quite confused about setting up this problem in FiPy.
a. Are these types of non-linear coefficients allowed for the transient-term ?
b. All the examples that I have seen thus far, use constant/scalar
coefficients. In my case, I have a coefficient-matrix which depends on the
spatial variable in the y-direction.
c. If coefficient-matrix is indeed allowed, what must be the data-type ?
The following is my general approach thus far,
* Construct a 'temporary' 1D mesh using the dy and ny parameters of the
original 2D mesh, and extract it's node-values using Cellcenters method. This
should correspond to the original nodes in the y-direction (am I right ?).
* Convert this to 'ndarray' datatype using numerix.ndarray.
* Transpose this, such as the result is a column vector.
The one-liner code that (potentially) implements this is shown below:
discretised_y_vector = numerix.array((Grid1D(dx = dy_2D_original , nx =
ny_2D_original)).cellCenters[0]).transpose()
Repeat (replicate) the column vector 'nx' times (in my case, 20 times) to
obtain the coefficient matrix for the transient term.
discretised_y_matrix = numerix.repeat(discretised_y_vector, nx)
Is this approach correct ? I am a beginner to FiPy and numerical PDE solving in
general. Any pointers in this direction shall be much appreciated.
Regards
Krishna
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