You are here

Numerical Methods for Conservation Laws

Jan S. Hesthaven
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
, on
A conservation law is an expression in mathematical terms of the balance within a physical system. It is a statement that the production of a physical quantity such as mass, energy or charge in a closed volume is exactly equal to the flux of that quantity across the boundary of that volume. Such conservation laws often take the form of partial differential equations with appropriate boundary conditions or equivalent integral forms.
The primary goal of this book is to provide a thorough introduction to the development of computational methods for conservation laws associated with hyperbolic partial differential equations (PDEs). A notable feature of such nonlinear PDEs is the formation in finite time of discontinuities in their solutions. These lead to well-known difficulties in getting accurate and efficient numerical solutions. The author provides a broad collection of powerful numerical techniques while establishing a solid basis for the underlying analysis.
While more basic monotone finite-difference schemes are presented here, the main topic is construction and analysis of high-order numerical methods and their use in practical applications. These methods include flux-limited methods, non-oscillatory schemes, spectral methods and some of the recently developed discontinuous Galerkin techniques.
MATLAB code to implement most of the methods is provided in the book and on the book’s website. (There are more than 150 MATLAB scripts here.) These include, for example, code for Maxwell’s equations, Burger’s equation, and Euler’s equation for gas dynamics and the transport equation. To keep the exposition focused and the software manageable, most of the examples are one-dimensional, but two-dimensional examples are described for a selection of methods. Throughout the book MATLAB code allows students to follow along to replicate the author’s computations and recreate the figures in the book.
The author uses a basic form of the Riemann problem as a prototype to introduce the major features of solutions of conservation laws. The starting point is a hyperbolic conservation law with piecewise constant initial conditions and a single discontinuity at the origin. In such a system properties such as shocks and rarefactions occur as characteristics in the solution.
The text is aimed at graduate students and researchers. It assumes a solid background in numerical analysis and a first course in partial differential equations. The range and variety of methods used here make considerable demands on the reader, but one who persists will learn much of what is currently known about hyperbolic conservation laws and their numerical solutions.
The book has a very complete bibliography and would provide excellent support for readers pursuing additional research. No exercises are provided but there are many examples worked out in detail.


Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and ceramic fiber-reinforced composites. Along the way he learned more about ceramics and alloys of aluminum than he could have imagined in graduate school. He did his PhD work in dynamical systems.