Numerical Differentiation help needed.
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:
(note: I didn't know how to show these as partial derivatives. All derivatives are partial here.)
Where F is a flux function
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,