Discretization of v*(du/dx)

Question

I am trying to discretize the term: $$\underline{v}\frac{d\underline{u}}{dx}$$ using finite differences or evaluate $$\int_{\Gamma}\underline{v}\frac{d\underline{u}}{dx}.\underline{n}d\Gamma$$ using finite volumes (in 1D). However, I have never come across a term where you have a variable multiplied by the divergence of another variable. Perhaps if there is a way to bring both u and v into the differential i.e. $\frac{d\underline{u}\underline{v}}{dx}$ but I am not sure if that is possible. I would appreciate any help. Thanks!

Answer 1

After checking your profile, it seems to me that this question is only one in a range of questions about how to solve numerically Navier-Stokes equations and the like. Therefore I've decided to provide instead some general information. Another reason is that your specific question cannot be answered easily as such. With Numerical Analysis, it is in general not possible to consider terms like $\;v\,\partial u / \partial x\;$ in isolation. I've been working for years in heat transfer and fluid flow. And did find satisfactory numerical analysis for related problems, but not for the NS equations themselves; the main reason for this being other priorities imposed by my employers. So for what it's worth, here is (part of) my work on it:

• Unified Numerical Analysis
• ZONWIND (Solar Wind)
• From Patankar's Book

There's a lot more on my web site (see profile), but not as systematical as it could be, unfortunately.

Apart from the above, an absolute must-have is the excellent book by Suhas Patankar . To my big surprise, it's completely on-line these days but I think it's more convenient to lay your hands on a printed paper instance (any decent library should have it):

• Numerical Heat Transfer and Fluid Flow
• the book at Amazon

As might be clear from the above, my own bias is towards a Unification of Finite Volume and Finite Element Methods. But it's not easy to find a good Finite Element text on fluid flow problems. I have been trying, instead, to incorporate the standard Finite Volume fluid flow treatment into a Finite Element (curvilinear) context, with some success.