# Implement Robin boundary condition (finite volume)

I have a PDE equation with Robin Boundary condition in an annulus system and I should solve it by finite volume method:

\begin{align} \frac{\partial T_f}{\partial t} - k \left(\frac{\partial^2 T_f}{\partial r^2} + \frac{1}{r} \frac{\partial T_f}{\partial r} \right) = -Q \end{align}

I do not have any problem in the discretization of inner points by finite volume method, but in $r_{ID}$ and $r_{OD}$, it has robin boundary condition which I do not know how to discretize it.

\begin{align} \left.\frac{\partial T_f}{\partial r}\right|_{r=r_{ID}} &= h(T_f-T_w ) \\ \left.\frac{\partial T_f}{\partial r}\right|_{r=r_{OD}} &= h(T_f-T_g ) \end{align}

(h&k are constant)
I would appreciate it if you could help me to discretize this boundary condition.

• When you test the $\frac{\partial^2 T_f}{\partial r^2}$ term, you do integration by parts, which produces a surface integral that includes $\frac{\partial T_f}{\partial r}$ as part of the integrand, right? Just substitute your Robin BCs into that term. If you detail your discretization process, it might be easier to give an explicit answer. May 14, 2017 at 16:54