I have been given a convection-diffusion ODE modeling the steady state temperature of a pipe (through which flows a fluid) as
$$-\frac{d}{dz}\left(\kappa \frac{dT}{dz} \right)+v\rho C\frac{dT}{dz}=Q(z)$$.
Here $T$ is the (steady state) temperature and it depends on the space variable $z$. All other parameters are constants and for the function $Q$ holds
$$Q(z)= \left\{\begin{matrix} 0,&\quad 0\leq z <a \\ Q_{0}\sin(\frac{(z-a)\pi}{b-a}),&\quad a\leq z \leq b \\ 0,&\quad b<z\leq L \end{matrix}\right.$$
As for the boundary conditions (BCs) we have a Dirichlet condition for the leftmost boundary and a Neumann condition for the rightmost boundary,
$$T(0)=T_{0}\quad \text{and}\quad -\kappa \frac{dT}{dz}(L)=\alpha(v)(T(L)-T_{out}),$$
where $T_{out}$ is the temperature outside the pipe to the right of $z=L$ and $\alpha$ is given by
$$\alpha(v) = \sqrt{\frac{v^{2}\rho^{2}C^{2}}{4}+\alpha_{0}^{2}}-\frac{v\rho C}{2}. $$
Now, I would like to discretize this ODE using a second order finite difference method (FDM), so I propose to use the central difference approximation which approximates the first and second derivatives as
$$\begin{align*} \frac{du}{dz}(z_{i})&=\frac{u(z_{i+1})-u(z_{i-1})}{2h} + O(h^{2})\\ \frac{d^{2}u}{dz^{2}}(z_{i})&=\frac{u(z_{i+1})-2u(z_{i})+u(z_{i-1})}{h^{2}} + O(h^{2}) \end{align*}$$
Finally, I would like to discretize the Neumann BC, but here things get tricky. If I use a central difference, I will obtain a point outside the domain (ghost point?). Could I use the condition that the temperature outside of the pipe at $z=L$ is $T_{out}$, or is this false? And also, how do I implement or force the Neumann condition to be fulfilled? Can I simply replace $dT/dz$ in the ODE with the BC when approximating the solution at $z=L$? This is what I do right now.
Interestingly enough, my solution seems to blow upp when I increase the number of steps, or mesh points, in the discretization of the $z$-domain. Look at these results:
The temperature completely sky rockets! Why is this? (If you would like me to post me code, I am more than willing to do so.)
Best regards,
