**Introduction**

This is an excellent introduction into
finite volume methods for solving conservation laws. The book is
divided into three main parts: Part I deals with linear equations
in predominately one spatial dimension, Part II introduces
nonlinear equations again in one spatial dimension, while Part III
introduces multidimensional problems. Beginning with an
introduction to the mathematics of these partial differential
equations, including all the basic hyperbolic theory for linear
equations (characteristics, Riemann problems, definition, and
examples), LeVeque then presents the basics of finite volume
methods for the integration of conservation laws. Extensions to
these basics in terms of high resolution methods such as the use
of limiters in multidimensional up-winding is then presented. The
nonlinear portion of the book begins with the mathematics of
nonlinear scalar conservation laws, the application of finite
volume methods for their numerical solution, extensions to systems
of equations, the nonlinear Riemann problem, non-classical
hyperbolic systems, and finally concludes with a chapter on
equations with source terms. The third part of the book deals
with multidimensional hyperbolic problems and numerical methods
both in scalar and vector form.

The book is exceptionally well written and the problems are worth
subsequent study. If you work in the area of hyperbolic systems
you must read this book.

One of the great bonuses that this book contains is that it
includes a description and introduction on the use of the
CLAWPACK
software. The CLAWPACK of codes is a large collection of FORTRAN 77 codes
for solving a large number of hyperbolic systems (both conservative
and nonconservative forms). Included is a "getting started" section
which can help new users become familiar with the codes and get them up and
solving problems very quickly.

I am in the process of transcribing my notes from this book into a solutions manual. You can find my first attempts at this process below. As always, please report any errors that may remain.

- My most recently typeset notes and problem solutions can be found here:

**Chapter 3 (Characteristics and Riemann Problmes for Linear Hyperbolic Equations):**- prob_3_1.m (phase plane solutions to linear Riemann problems)

- riemann2x2.m (solves a 2x2 linear Riemann problem)

- prob_3_3_b.m (plots the x-t solution for this linear Riemann problem)

- prob_3_4.m (plots the region in x-t plane where solution for this linear Riemann change)

- prob_3_1.m (phase plane solutions to linear Riemann problems)
**Chapter 5 (Introduction to the CLAWPACK software):**- Problem 5.1 (visualization of the solution to a 1-d advection flow problem with the upwind method)

- Problem 5.2 (the upwind method with Courant number larger than 1)

- Problem 5.3 (a pure left going pressure pulse)

- Problem 5.1 (visualization of the solution to a 1-d advection flow problem with the upwind method)
**Chapter 9 (Variable-Coefficient Linear Equations):****Chapter 13 (Nonlinear Systems of Conservation Laws):**- The Shallow Water Equations:
- prob_13_2.m (rarefaction wave curves for Example 13.5 plotted in the h-u plane)

- prob_13_4.m (phase plane solution of a one-shock/two-shock collision)

- prob_13_2.m (rarefaction wave curves for Example 13.5 plotted in the h-u plane)
- The p-System:
- prob_13_7_d.m (the Hugoniot loci for various equation of states)

- prob_13_7_e.m (a Riemann problem with a two-shock solution)

- prob_13_7_d.m (the Hugoniot loci for various equation of states)
- prob_13_11_b.m (plots the Hugoniot/Integral curves for Problem 13.11 Part b)

- prob_13_11_c.m (plots the phase plane solution for an example Riemann problem)

- prob_13_11_d.m (plots the phase plane solution for Riemann problem with no solution)

- prob_13_12.m (plots the solution for the Riemann problem in the x-t plane)

- The Shallow Water Equations:
**Chapter 16 (Some Nonclassical Hyperbolic Problems):**- prob_16_1_pt_a.m (plots the x-t figure for Problem 16.1 Part a)

- prob_16_1_pt_a.m (plots the x-t figure for Problem 16.1 Part a)

John Weatherwax Last modified: Thu Jan 11 13:26:54 EST 2007