Diffusion Advection equation discretization scheme I am looking for a good reference to understand the basic discretization schemes applied to the Stationary Diffusion Advection equation.
$$-\epsilon \frac{d^2u(x)}{dx^2}+\beta \frac{du(x)}{dx}=0$$
Especially for the Upwind and the Scharfetter-Gummel schemes.
Thank you in advance.
 A: Just from looking at your equation, I would guess that you can use just about any basic discretization scheme. Schemes like Upwind and Scharfetter-Gummel (I had to look up this last one) were developed to handle partial differential equations that include dependence on time, which can introduce all sorts of numerical headaches. 
If you need a general guide on numerical methods for ODE's, there are a number of books that would work. If you want free and instant information, I'd recommend this Wikipedia article. It discusses three basic schemes in detail and then suggests further reading for more advanced methods. 
If you are already comfortable with this introductory material and are looking for a more accurate scheme, I would personally use one of the Runge-Kutta methods, probably the popular 4th order one because it is the one with which I am most comfortable. These methods were developed to solve first-order equations and so you need to do it twice (once on $u'$ and again on $u''$). Some Mathematica code to do exactly this, which you can use as pseudocode if you are using some other software / language, can be found here. 
If you want to know about Upwind, I think the Wikipedia article on this is quite usable. Here is a pdf on Sharfetter-Gummel. I want to emphasize again that I am pretty sure these methods are for partial differential equations. At the very least, their general descriptions in these sources are formulated at such. I do not know if they can be usefully translated to ODE methods (by 'useful' I mean without erasing their purpose). 
Lastly, this is a publication presenting a numerical solution to a diffusion-advection system that appears a little more advanced than the preceding.
