I'm looking at the 1D Advection-Diffusion Equation $$\frac{\partial\phi}{\partial t}+\frac{\partial}{\partial x}\left(u\phi - D\frac{\partial \phi}{\partial x}\right)=0$$ with $u$ and $D$ constant and the following Dirichlet-Neumann Boundary Conditions: $$\phi(t,0)=\alpha,\quad\text{ and }\quad\phi_{x}(t,1)=\beta$$ I have a very similar case, but for simplicity we can consider with the 2nd Order CDS Scheme $$ u\frac{\phi_{i+1}-\phi_{i-1}}{2 \Delta x}-D\frac{\phi_{i+1}-2\phi_{i}+\phi_{i-1}}{\Delta x^{2}}=0$$ Does Neumann stability analysis still hold? I have not been able to find the explanation out right if why. Also does the inhomogeneous case differ?
ADDITIONAL INFORMATION:
Take $P=\frac{u\Delta x}{D}$ s.t. the 2nd Order CDS scheme becomes $$ -\frac{D}{\Delta x^{2}}\left[\left(1-\frac{P}{2}\right)\phi_{i+1}-2\phi_{i}+\left(1+\frac{P}{2}\right)\phi_{i-1}\right]=0$$ where the discretization at the boundaries is done by:
Left Dirichlet BC: $\alpha = \frac{\phi_{1}+\phi_{0}}{2}\implies \phi_{0}=-\phi_{1}+2\alpha$
such that $$-\frac{D}{\Delta x^{2}}\left[\left(1-\frac{P}{2}\right)\phi_{2}-\left(3+\frac{P}{2}\right)\phi_{1}\right]=2\frac{D}{\Delta x^{2}}\left(1+\frac{P}{2}\right)\alpha$$
Right Neumann BC: $\beta = \frac{\phi_{N+1}+\phi_{N}}{\Delta x}\implies \phi_{N+1}=-\phi_{N}+\Delta x\beta$
such that
$$-\frac{D}{\Delta x^{2}}\left[\left(\frac{P}{2}-3\right)\phi_{N}+\left(1+\frac{P}{2}\right)\phi_{N-1}\right]=\frac{D}{\Delta x}\left(1-\frac{P}{2}\right)\beta$$
How would you show that Fourier Modes don't apply to the boundaries?