I have had a class on numerical methods in which we learned the basics of boundary value problems, Lax equivalence theorem, consistency, stability, and a small amount of PDE stuff (mostly just Von Neumann stability analysis, and second order (centered) finite difference approximations for linear hyperbolic equations.
I want to simulate wave equations, both linear and nonlinear. However the more I look into hyperbolic PDEs, the more I realize that I do not know enough to continue.
From online notes, I learned about the CFL number, that numerical dispersion and dissipation exist , and that it is possible for a method to cause a phase shift (I don't know how to estimate any of these quantities). I have also heard of the method of lines, but my only attempt produced an approximation that quickly went to zero.
I would like to find a resource that I can study carefully, with examples of the above. I would like to learn about alternating implicate methods (I will need an implicate method to study waves with a (physical) dispersion relation), the method of lines and how to analyse dispersion/dissipation/phase problems. Higher-order methods would be a bonus.
Online notes are alright, but they tend to be rather incomplete. I would really like to get a full textbook on the subject. Something specializing in hyperbolic PDEs would be nice, but I am not against more general books.
I would like to know people's favorites, and the reasons that they are thought to be good books.