The basic class of finite volume methods developed in this book has been implemented in the software package CLAWPACK. This allows these algorithms to be applied to a wide variety of hyperbolic systems simply by providing the appropriate Riemann solver, along with initial data and boundary conditions. The high-resolution methods introduced in Chapter 6 are implemented, but the simple first-order Godunov method of Chapter 4 is obtained as a special case by setting the input parameters appropriately. (Specifically, set method(2)=1 as described below.) In this chapter an overview of the software is given along with examples of its application to simple problems of advection and acoustics.

The software includes more advanced features that will be introduced later in the book, and can solve linear and nonlinear problems in one, two, and three space dimensions, as well as allowing the specification of capacity functions introduced in Section 2.4 (see Section 6.16) and source terms (see Chapter 17). CLAWPACK is used throughout the book to illustrate the implementation and behavior of various algorithms and their application on different physical systems. Nearly all the computational results presented have been obtained using CLAWPACK with programs that can be downloaded to reproduce these results or investigate the problems further. These samples also provide templates that can be adapted to solve other problems. See Section 1.5 for details on how to access webpages for each example.

Many multidimensional problems of practical interest involve complex geometry, and in general it is not sufficient to be able to solve hyperbolic equations on a uniform Cartesian grid in a rectangular domain. In Section 6.17 we considered a nonuniform grid in one space dimension and sawhowhyperbolic equations can be solved on such a grid by using a uniform grid in computational space together with a coordinate mapping and appropriate scaling of the flux differences using capacity form differencing. The capacity of the computational cell is determined by the size of the corresponding physical cell.

In this chapter we consider nonuniform finite volume grids in two dimensions, such as those shown in Figure 23.1, and will see that similar techniques may be used. There are various ways to view the derivation of finite volume methods on general multidimensional grids. Here we will consider a direct physical interpretation in terms of fluxes normal to the cell edges. For simplicity we restrict attention to two space dimensions. For some other discussions of finite volume methods on general grids, see for example.

The grids shown in Figures 23.1(a) and (b) are logically rectangular quadrilateral grids, and we will concentrate on this case. Each cell is a quadrilateral bounded by four linear segments. Such a grid is also often called a curvilinear grid.

Whenever we use a numerical method to solve a differential equation, we should be concerned about the accuracy and convergence properties of the method. In practice we must apply the method on some particular discrete grid with a finite number of points, and we wish to ensure that the numerical solution obtained is a sufficiently good approximation to the true solution. For real problems we generally do not have the true solution to compare against, and we must rely on some combination of the following techniques to gain confidence in our numerical results:

• Validation on test 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 estimates for the numerical error that will be observed on any particular finite grid.

In this chapter we concentrate on the theoretical analysis. Here we consider only the Cauchy problem on the unbounded spatial domain, since the introduction of boundary conditions leads to a whole new set of difficulties in analyzing the methods. We will generally assume that the initial data has compact support, meaning that it is nonzero only over some bounded region.