Hyperbolic conservation laws describe a number of interesting physical problems
in diverse areas such as fluid dynamics, solid mechanics, and astrophysics. Our
emphasis in this book is on nonlinearities in these problems, especially those that
lead to the development of propagating discontinuities. These propagating discontinuities can appear as the familiar shock waves in gases (the “boom” from explosions
or super-sonic airplanes), but share many mathematical properties with other waves
that do not appear to be so “shocking” (such as steep changes in oil saturations in
petroleum reservoirs). These nonlinearities require special treatment, usually by
methods that are themselves nonlinear. Of course, the numerical methods in this
book can be used to solve linear hyperbolic conservation laws, but our methods will
not be as fast or accurate as possible for these problems. If you are only interested
in linear hyperbolic conservation laws, you should read about spectral methods and
multipole expansions.
This book grew out of a one-semester course I have taught at Duke University
over the past decade. Quite frankly, it has taken me at least 10 years to develop the
material into a form that I like. I may tinker with the material more in the future,
because I expect that I will never be fully satisfied.
I have designed this book to describe both numerical methods and their applications. As a result, I have included substantial discussion about the analytical solution
of hyperbolic conservation laws, as well as discussion about numerical methods. In
this course, I have tried to cover the applications in such a way that the engineering
students can see the mathematical structure that is common to all of these problem
areas. With this information, I hope that they will be able to adapt new numerical
methods developed for other problem areas to their own applications. I try to get the
mathematics students to adopt one of the physical models for their computations
during the semester, so that the numerical experiments can help them to develop
physical intuition.