I have the following PDE, with parameters $a$ and $b$:
$$ \frac{\partial c}{\partial t} = \frac{\partial}{\partial z} \left( a c + b \frac{\partial c}{\partial z}\right) $$ with, for now, just one boundary condition $$ \left.\left( a c + b \frac{\partial c}{\partial z}\right)\right|_{z=0} = 0 $$
I'm solving this PDE numerically using FD. I'm implementing another boundary condition when solving it numerically: $$ c = 0 ~ \text{at} ~ z=H $$ where $H\gg h$. The initial condition is a linear profile with negative gradient between $0$ and $h$ (i.e. $c=0$ at $z=h$ and $c=c_b$ at $z=0$)
For certain relation between the parameters $a$ and $b$ I obtain the transient solution.
$$ c(z) = A ~\text{exp}\left(\frac{-az}{b} \right) $$
I cannot use the boundary condition I implemented in the numerical solution to obtain the constant $A$ because $c$ is never $0$ in the steady-state solution. However, in the numerical solution, I can get a steady state solution for certain values of $a$ and $b$, whereas for other values, the solution grows indefinitely (probably because of the indefinition of the constant A). Why is this happening? What are the conditions for stability or instability? I know I could just set an additional boundary condition and make it stable, I just want to know why the numerical scheme is obtaining the steady-state solution for certain values of $a$ and $b$. Thanks