In the previous chapters on nonlinear problems we have concentrated on classical systems of conservation laws for which the wave structure is relatively simple. In particular, we have assumed that the system is strictly hyperbolic (so that there are m distinct integral curves through each point of phase space), and that each characteristic field is either linearly degenerate or genuinely nonlinear (so that the eigenvalue is constant or varies monotonically along each integral curve). Many important systems of equations satisfy these conditions, including the shallow water equations and the Euler equations for an ideal gas, as well as linear systems such as acoustics. However, there are other important applications where one or both of these conditions fail to hold, including some problems arising in nonlinear elasticity, porous-media flow, phase transition, and magnetohydrodynamics (MHD). In this chapter we explore a few of the issues that can arise with more general systems. This is only an introduction to some of the difficulties, aimed primarily at explaining why the above assumptions lead to simplifications.

We start by considering scalar conservation laws with nonconvex flux functions (which fail to be genuinely nonlinear because f″(q) vanishes at one or more points). This gives a good indication of the complications that arise also in systems of more equations that fail to be genuinely nonlinear. Then in Section 16.2 we will investigate the complications that can arise if a system is not strictly hyperbolic, i.e., if some of the wave speeds coincide at one or more points in phase space.

In this chapter we begin to study finite volume methods for the solution of conservation laws and hyperbolic systems. The fundamental concepts will be introduced, and then we will focus on first-order accurate methods for linear equations, in particular the upwind method for advection and for hyperbolic systems. This is the linear version of Godunov's method, which is the fundamental starting point for methods for nonlinear conservation laws, discussed beginning in Chapter 15. These methods are based on the solution to Riemann problems as discussed in the previous chapter for linear systems.

Finite volume methods are closely related to finite difference methods, and a finite volume method can often be interpreted directly as a finite difference approximation to the differential equation. However, finite volume methods are derived on the basis of the integral form of the conservation law, a starting point that turns out to have many advantages.

General Formulation for Conservation Laws

In one space dimension, a finite volume method is based on subdividing the spatial domain into intervals (the “finite volumes,” also called grid cells) and keeping track of an approximation to the integral of q over each of these volumes. In each time step we update these values using approximations to the flux through the endpoints of the intervals.

A brief introduction to one-dimensional elasticity theory and elastic wave propagation was given in Section 2.12. In this chapter we will explore the full three-dimensional elasticity equations in the context of elastic wave propagation, or elastodynamics. There are many references available on the basic theory of linear and nonlinear elastodynamics (e.g., though often not in the first-order hyperbolic form we need. In this chapter the equations, eigenstructure, and Riemann solutions are written out in detail for several different variants of the linear problem.

The notation and terminology for these equations differs widely between different fields of application. Much of the emphasis in the literature is on steady-state problems, or elastostatics, in which the goal is to determine the deformation of an object and the internal stresses that result from some applied force. These boundary-value problems are often posed as second-order or fourth-order elliptic equations. We will concentrate instead on the hyperbolic nature of the first-order time-dependent problem, and the eigenstructure of this system. This is important in many wave-propagation applications such as seismic modeling in the earth or the study of ultrasound waves propagating through biological tissue. For small deformations, linear elasticity can generally be used. But even this case can be challenging numerically, since most practical problems involve heterogeneous materials and complicated geometry. High-resolution finite volume methods are well suited to these problems, since interfaces between different materials are handled naturally in the process of solving Riemann problems.

In Chapter 3 we developed the theory of linear systems of hyperbolic equations, in which case the general Riemann problem can be solved by decomposing the jump in states into eigenvectors of the coefficient matrix. Each eigenvector corresponds to a wave traveling at one of the characteristic speeds of the system, which are given by the corresponding eigenvalues of the coefficient matrix.

In Chapter 11 we explored nonlinear scalar problems, and saw that when the wave speed depends on the solution, then waves do not propagate unchanged, but in general will deform as compression waves or expansion waves, and that shock waves can form from smooth initial data. The solution to the Riemann problem (in the simplest case where the flux function is convex) then consists of a single shock wave or centered rarefaction wave.

In this chapter we will see that these two theories can be melded together into an elegant general theory for nonlinear systems of equations. As in the linear case, solving the Riemann problem for a system of m equations will typically require splitting the jump in states into m separate waves. Each of these waves, however, can now be a shock wave or a centered rarefaction wave.

We will develop this general theory using the one-dimensional shallow water equations as a concrete example. The same theory will later be illustrated for several other systems of equations, including the Euler equations of gas dynamics in Chapter 14. The shallow water equations are a nice example to consider first, for several reasons. It is a system of only two equations, and hence the simplest step up from the scalar case.

In this chapter the high-resolution wave-propagation algorithms developed in Chapter 20 for scalar problems are extended to hyperbolic systems. We start with constant-coefficient linear systems, where the essential ingredients are most easily seen. A Riemann problem is first solved normal to each cell edge (a simple eigendecomposition in the linear case). The resulting waves are used to update cell averages on either side. The addition of correction terms using wave limiters (just as in one dimension) gives high-resolution terms modeling the pure x- and y-derivative terms in the Taylor series expansion (19.5). The crossderivative terms are handled by simple extension of the corner-transport upwind (CTU) idea presented for the advection equation in Sections 20.2 through 20.5. In general this requires solving a second set of Riemann problems transverse to the interface. For a linear system this means performing a second eigendecomposition using the coefficient matrix in the transverse direction. Extending the methods to variable-coefficient or nonlinear systems is then easy, using ideas that are already familiar from one space dimension. The solutions (or approximate solutions) to the more general Riemann problems are used in place of the eigendecompositions, and the method is implemented in a wave-propagation form that applies very generally.

Constant-Coefficient Linear Systems

We again consider the constant-coefficient linear system qt + Aqx + Bqy = 0 discussed in Chapter 19, where in particular the Lax–Wendroff and Godunov methods for this system were presented. The numerical fluxes for these two methods are given by (19.14) and (19.18) respectively.

Hyperbolic partial differential equations arise in a broad spectrum of disciplines where wave motion or advective transport is important: gas dynamics, acoustics, elastodynamics, optics, geophysics, and biomechanics, to name but a few. This book is intended to serve as an introduction to both the theory and the practical use of high-resolution finite volume methods for hyperbolic problems. These methods have proved to be extremely useful in modeling a broad set of phenomena, and I believe that there is need for a book introducing them in a general framework that is accessible to students and researchers in many different disciplines.

Historically, many of the fundamental ideas were first developed for the special case of compressible gas dynamics (the Euler equations), for applications in aerodynamics, astrophysics, detonation waves, and related fields where shock waves arise. The study of simpler equations such as the advection equation, Burgers’ equation, and the shallow water equations has played an important role in the development of these methods, but often only as model problems, the ultimate goal being application to the Euler equations. This orientation is still reflected in many of the texts on these methods. Of course the Euler equations remain an extremely important application, and are presented and studied in this book, but there are also many other applications where challenging problems can be successfully tackled by understanding the basic ideas of high-resolution finite volume methods. Often it is not necessary to understand the Euler equations in order to do so, and the complexity and peculiarities of this particular system may obscure the more basic ideas.