Numerical Differentiation help needed.

Hello,

I have a bit of a problem that I was hoping someone might be able to help me with. I'll start by explaining what I am attempting to do and explain where I need help.

I'm trying to program a simulation of the Viscous Burger's Equation in 1 dimension. I'm going to use an implicit finite difference scheme (Crank-Nicolson) to do this.

The problem I have is the fact that the Burger's equation I mentioned is non-linear, I need to have this equation in conservation form before I apply my scheme. The Burgers equation in conservation form is as follows:

$\displaystyle dU/dt + dF/dx = vd^2U/dx^2$ (note: I didn't know how to show these as partial derivatives. All derivatives are partial here.)

Where F is a flux function $\displaystyle F = U^2/2$

and v is the viscosity which is a constant.

I was just wondering what is the best way of linearizing this equation? If you can show me or give me steps to follow so I can ask follow up questions that would be brilliant.

Is it best to linearize this before or after applying the difference scheme? (which is nicer?)

I want to thank you for any help you can give in advance.

All the best,

Wayne.