# Finite differences linear PDE

Suppose you want to write a finite difference scheme for a linear PDE of the form

$V_t + x V_x + x^2 V_{xx} =0$,

where the subscripts x,t denote derivatives with respect to space and time respectively. I know that using Taylor expansions, one can derive forward/backward or theta schemes for $V_t$, $V_x$ and $V_{xx}$. However, how do I deal with the terms $x V_x$ and $x^2 V_{xx}$ ? Is it okay (in terms of convergence and stability) to e.g. approximate $x V_x(t,x_0)$ by something like $x_0 (\frac{V(t,x_1)-V(t,x_0)}{x_1-x_0})$?

I've searched the web but I can't find any useful links. Even some useful keywords would be very much appreciated.

• This is a linear PDE. – Artem Jan 18 '14 at 0:00
• Thanks. Obviously I thought that (non)linear refers to the coefficients of the PDE, but it refers to the dependent variable V and its derivatives only. – Stochast Jan 18 '14 at 10:28