Introductory text to Peter Lax Theorem

Question

I'm looking for Numerical Analysis's books that explains about Finite Difference, Convergence, Consistency and Estability in a introductory level, moreover I want see some applications of Peter Lax Theorem in PDE's.

PS: My background in Numerical Analysis is a null set.

Answer 1

I would recommend "Finite Difference Methods for Ordinary and Partial Differential Equation: Steady state and time-dependent problems", by Randall J. LeVeque. In the second chapter he shows how to prove convergence of finite difference schemes that are consistent and stable at a fairly introductory level

The back of the book actually has an endorsement by Peter Lax himself!