Hello,
I am trying to solve a bunch of advection-diffusion-reaction
equations for $N$ scalar fields coupled to a flow field in
$d=1,2$ dimensions.
Equation for the vectorial flow field
$$
\nabla^2 \mathbf{v} + \zeta \nabla (\nabla \cdot \mathbf{v})
= \mathbf{v} - \nabla F(\{c_i\})
$$
The flow field is not divergence free. $F$ is a function of
the $N$ scalar fields $c_i$, $i=1,\ldots N$. These scalar
fields are evolved by coupled advection-diffusion-reaction
equations of the form
$$
\frac{\partial c_i}{\partial t} =
-\mathbf{v} \cdot \nabla c_i
-(\nabla \cdot \mathbf{v}) c_i
+ D_i \nabla^2 c_i + R_i(\{c_i\})
$$
where $D_i$ are the diffusion constants and $R_i$ are
(polynomial) reaction terms.
------------------
Before investing a lot of time, I want to know if this
system of equations can be solved via FIPY in $d=1,2$ in flat
and curved geometries (e.g., on a circular/elliptical loop
for 1D, on a spherical/ellipsoidal surface for 2D). The
example of the Cahn-Hilliard equation solved on a sphere is
very promising.
For 2D, this post
http://thread.gmane.org/gmane.comp.python.fipy/2819/focus=2833
might be relevant, and I dont know if one can still solve the
equations despite the discussion in that thread.
I started with the simplest case of a single scalar field on
a flat line in 1D, with no reaction terms. For a given $c$
field, I can get the flow field and for a given flow field,
I can evolve the advection-diffusion-reaction equation for
the $c$ field. The trouble is when they are coupled and
evolved together.
My attempt is here
http://pastebin.com/vrnTX5sj
Might be a silly mistake which I am missing/dont-understand.
Any help would be much appreciated.
Also could you please give your opinion regarding 2D and
curved geometries?
--
With my best regards
Vijay
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