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 applied to eliminate numerical oscillations. The methods were orginally designed to capture shock waves accurately, but are also useful tools for studying linear wave-progagation problems, particulary in heterogenous material. The methods studied are in the CLAWPACK software package. Source code for all the examples presented can be found on the web, along with animations of many of the simulations. This provides an excellent learning environment for understanding wave propagation phenomena and finite volume methods.

Hirsch [198], Laney [256], Oran & Boris [348], Peyret & Taylor [359], and Tannehill, Anderson & Pletcher [445]. These books discuss the ﬂuid dynamics in more detail, generally with emphasis on speciﬁc applications. For an excellent collection of photographs illustrating a wide variety of interesting ﬂuid dynamics, including shock waves, Van Dyke’s Album of Fluid Motion [463] is highly recommended. Many more references on these topics can easily be found these days by searching on the web. In

ﬂux may depend on the values of any or all of the conserved quantities at that point. Again it should be stressed that this differential form of the conservation law is derived under the assumption that q is smooth, from the more fundamental integral form. Note that when q is smooth, we can also rewrite (2.39) as qt + f (q)qx = 0, (2.41) where f (q) is the Jacobian matrix with (i, j) entry given by ∂ f i /∂q j . The form (2.41) is called the quasilinear form of the equation, because it

set of hyperbolic equations that must be solved for all motions of the solid, which are coupled together. However, if we restrict our attention to one-dimensional plane waves, in which all quantities vary only in one direction, then these equations decouple into two independent hyperbolic systems of two equations each. Mathematically these linear systems are not very interesting, since each has the same structure as the acoustics equations we have already studied in detail. Because of this,

variable CLAW in Unix so that the proper ﬁles can be found: unix> setenv CLAW /claw You might want to put this line in your .cshrc ﬁle so it will automatically be executed when you log in or create a new window. Now you can refer to $CLAW/clawpack/1d, for example, and reach the correct directory. 5.3.3 Compiling the code Go to the directory claw/clawpack/1d/example1. There is a ﬁle in this directory named compile, which should be executable so that you can type unix> compile This should

problems. The method (and particular implementation) should be tested on simpler problems for which the true solution is known, or on problems for which a highly accurate comparison solution can be computed by other means. In some cases experimental results may also be available for comparison. • Theoretical analysis of convergence and accuracy. Ideally one would like to prove that the method being used converges to the correct solution as the grid is reﬁned, and also obtain reasonable error