Finite Volume Methods for Hyperbolic Problems

Randall J. LeVeque

Published by Cambridge University Press in 2002.

This is a revised and expanded version of
Numerical Methods for Conservation Laws, ETH Lecture Notes, Birkhauser-Verlag, Basel, 1990.
Print: ISBN 3-7643-2464-3.   Online: ISBN 978-3-0348-5116-9


This book contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, including both linear problems and nonlinear conservation laws. These equations describe a wide range of wave-propagation and transport phenomena arising in nearly every scientific and engineering discipline. Several applications are described in a self-contained manner, along with much of the mathematical theory of hyperbolic problems. High-resolution versions of Godunov's method are developed, in which Riemann problems are solved to determine the local wave structure and limiters are then applied to eliminate numerical oscillations. These methods were originally designed to capture shock waves accurately, but are also useful tools for studying linear wave-propagation problems, particularly in heterogenous material. The methods studied are implemented in the Clawpack software. software package and source code for all the examples presented can be found on the web, along with animations of many time-dependent solutions. This provides an excellent learning environment for understanding wave-propagation phenomena and finite volume methods. Table of Contents and Introduction

Additional Materials to Accompany the Book

Simulations and Clawpack Software:

Most of the figures in the book showing computational results were created using Clawpack Version 4.3. Many of the examples have since been converted to the newer Clawpack Version 5.x form, and can be viewed in the Gallery of fvmbook applications, while the source code is located in the apps repository, in the fvmbook directory, which you can view on GitHub. For more information, see also the documentation page Examples from the book FVMHP.

New book:

The book Riemann Problems and Jupyter Solutions by D.I. Ketcheson, R.J. LeVeque, and M.J. del Razo (SIAM, 2020) illustrates many aspects of Riemann problems and their solution in the form of Jupyter notebooks.

Github Repository:

Additional materials described below are collected in the Github repository clawpack/fvmhp_materials. This material has a CC-BY license and can be freely used and altered for your own purposes (with attribution), see LICENSE.txt.

Videos and slides:

A series of 25 lectures recorded in 2023, originally as supplementary material for the graduate level course Applied Math 574 at the University of Washington, Videos are available on the Clawpack YouTube channel in the FVMHP playlist.

Slides to accompany these lectures (both pdf and the latex source) are available on GitHub in the clawpack/fvmhp_materials repository.

See slides_pdf/ for a brief list of the contents of each slide deck, and this page from AMath 574 for more detailed contents of each lecture and links to individual videos and slide decks.

See slides/ for instructions on how to use the latex files to create your own modified slide decks.

Jupyter Notebooks:

Eventually some Jupyter notebooks will be added to this repository. A few are available in .