Partial differential equation discretization. HELP D:

[maistral]

So figuratively, I'm trying to win a nuclear war with a stick. I did not take any course in PDEs, I just self-studied some of them, and now I'm toast.

First, please feel free to hurl rocks at me if my simplification is incorrect:

https://fbcdn-sphotos-g-a.akamaihd.net/hphotos-ak-prn2/t1/q71/1545789_719161698116788_1463986171_n.jpg

Second, how do you discretize ∂²u/(∂x∂y)? Is it:

(u_x+1,y - 2u_x,y + u_x-1,y)(u_x,y+1 - 2u_x,y + u_x,y-1)/((x_i+1 - x_i)(y_i+1 - y_i))

I might have a few more questions after this. Sorry if everything I'm saying is incorrect, please don't be harsh. I'm telling the truth when I say that we didn't cover this in our numerical methods class and I didn't take a PDE course. Thanks!

 

[maka89]

First:
Your equations looks right as long as $\mu$ does not depend on x…
(If $\mu$ does depend on x, they might still be a decent approximation if $\mu$ does not vary much, but I don't know what problem you are working on... )

Second:

Last section of: http://en.wikipedia.org/wiki/Finite_difference . ;)

I don't have any PDE courses either, so not sure exactly how to derive it. I think you can derive it straight from the definition of the taylor series of a function of two variables and dropping higher order terms :)