I am new to numerical methods for PDEs, but I am seeing some confusing perspectives in two different common textbooks: Langtangen's book on Finite Differences and Leveque's book on Finite Volume Methods. I was hoping someone could help clarify.
I understand the difference between finite difference methods and finite volume methods. Finite difference use the value of the numerical derivatives at fixed points while finite volumn methods use the value of the control volumes between fixed interface points. The claim is generally that finite volume methods can preserve conservation laws better than finite differences.
The confusing thing is that I am seeing the same methods referenced in both books on finite differences and finite volume. In other words, I am seeing upwind scheme and Lax-Wendroff methods referenced in Langtangen's book for finite differences, and also in Leveque's book on finite volume methods. I am including snapshots of the relevant table of contexts sections for both books to show the overlap.
How can the same scheme belong to two different methods? Or am I misinterpreting something. Here are the excerpt from the tables of contents below.
Langtangen:
Leveque:
