[SOLVED] How to represent these non-standard pdes with FiPy?

How to represent these non-standard pdes with FiPy?

I am trying to solve a set of PDEs with FiPy, but not sure how to represent the equations: see the PDEs here.

The terms give me trouble are those in red rectangle and blue rectangle. The one in red rectangle is a convection term times a function depending on x, T1 and T2; The one in blue rectangle is a transient term times another function depending on x, T1 and T2.

I couldn't find any example including this kind of terms. Could you please give me some suggestions, or share me an example?

I appreciate any help. Thanks in advance.

Solution

This question is similar to cellvariable*Diffusion in fipy, but for convection instead of diffusion. The solution is the same though. Use,

$f \partial_x \phi = \partial_x \left( f \phi \right) - \phi \partial_x f$

to convert the red terms into a convection term with a coefficient of f and a source term.

For the blue terms do exactly the same to get a transient term with a coefficient and a source term.

Edit: if we assume that $\inline f = f\left( g \left( x \right ), \phi \left( x \right) \right )$ then we can still use

$f \phi_x = \left( f \phi \right)_x - \phi f_x$

and approximate explicitly (with f as a variable in FiPy and use grad). However, we can take it further and use,

$f \phi_x = \left( f \phi \right)_x - \phi f_g g_x - \phi f_{\phi} \phi_x$

and approximate the final $\inline \phi_x$ explicitly.

Again, we can go even further with,

$f \phi_x = \left[ \left(f - \frac{1}{2} f_{\phi} \right) \phi \right]_x - \phi \left[f_g g_x - \frac{1}{2} \phi f_{\phi g} g_x\right] + \frac{1}{2}\phi^2 f_{\phi \phi} \phi_x$

Again the last term can be solved explicitly. Depending on the form of f the last them should become more insignificant and thus, the explicitness less of an issue.