Resource request: Numerical Methods for Hyperbolic PDE

Question

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.

Answer 1

It is a rather wide spectrum of topics you for which you request references. Though there certainly is literature available that attempts to cover both the theory of (hyperbolic) PDEs as well as their discretisation in the same text, in my experience they rarely do a very good job.

Since your experience with numerical methods for PDEs are at an introductory level, my strong recommendation for a first source is High order difference methods for time dependent PDE by Bertil Gustafsson. It is a relatively easy read, suitable as an introductory text to the more advanced aspects of PDE discretisations with a focus on finite difference methods. It touches upon topics such as explicit/implicit methods, CFL, the method of lines, analytic and numerical dispersion relations and high order methods. In particular, it includes a chapter on Summation-by-Parts operators and stability beyond von Neumann (which will be absolutely crucial when you start to handle boundary conditions and/or non-linear problems). It does not cover a very deep mathematical analysis; it just scrapes the surface of each topic deep enough to give an understanding of its importance and how to handle its intricacies.

Now, when it comes to hyperbolic PDEs, some mathematical depth is necessary. We cannot expect to obtain a reliable (or even sensible) numerical solution without some understanding of the problem at hand. In order to have any chance at tackling a hyperbolic initial-boundary value problem, an understanding of the concept(s) of well-posedness is necessary; which often boils down to an understanding of boundary conditions. My go-to texts are Time dependent problems and difference methods by Gustafsson, Kreiss and Oliger, as well as Initial-boundary value problems and the Navier-Stokes equations by Kreiss and Lorenz. They cover the theory of hyperbolic and parabolic initial-, boundary- and initial-boundary value problems in depth. The major advantage with these books is that there is a crystal clear connection between the continuous side (the PDE) and the discrete side (the numerical approximation) of the problem, in particular when it comes to well-posedness (PDE) and stability (approximation) and boundary conditions (PDE and approximation). These texts are somewhat more demanding however, relying on some familiarity with functional analysis.

The authors of the above books were all central figures in the development of the modern theory of stability and convergence of numerical schemes in the 70s, 80s and 90s. I think they will do a good job at picking you up where the introductory courses leave you, and bring you to a level of understanding where you can solve some really interesting problems.

Answer 2

For people interested by the theory and the numerical resolution of hyperbolic systems of conservation laws (i.e. first-order hyperbolic PDEs of the type ${\bf u}_t + {\bf f}({\bf u})_x = {\bf s}$), this topic has its own literature. Even if there must be much more textbooks than in the following list, it includes what can be considered as reference books on the topic, which all have their specificity.

The following books could be suggested for people interested in physical understanding, numerical methods, and everyday applications:

• [1] R.J. LeVeque: Numerical Methods for Conservation Laws, 2nd ed., Birkhäuser, 1992. Based on lecture notes, this book introduces the main theoretical and numerical concepts in the scalar case. There are many exercises and ready-to-compute examples. This book is recommended for beginner readers, who want to get used to the concepts that arise in such problems. One can try clicking the link (figure in brackets) to make his own opinion.
• [2] R.J. LeVeque: Finite-Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. This book is much more complete than the previous one, but is written in a slightly more condensed fashion.
• [3] E. Godlewski, P.-A. Raviart: Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996. This book is an overview of the topic. The presentation has less illustrations, but is very detailed.

The next books focus on the theoretical side:

• [4] J. Smoller: Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer, 1994. This book introduces many mathematical tools for the analysis/resolution of problems involving conservation laws, but not only.
• [5] C.M. Dafermos: Hyberbolic Conservation Laws in Continuum Physics, 2nd ed., Springer, 2005. This book focuses on the theory of systems of conservation laws.