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.

Domain decomposition for numerical solution of PDEs has been an active area of research, with a well-organized international conference series held annually since 1987. The area encompasses preconditioning of linear systems, discretizations, and solution of hybrid systems (e.g., coupled Navier–Stokes and Euler problems). We consider the first two of these in the context of high-order methods.

Introduction

Domain decomposition (dd) alleviates the solution complexity associated with the full problem in a complicated geometry. Broadly, the aim of dd consists in formulating independent problems in separate subdomains whose union constitutes the whole. The decomposition of the domain may be motivated by differing physics within different subdomains, by the availability of fast solvers for each subdomain, by the desire to partition the computational effort across separate processors, or by the inherent heterogeneity of the discretization. We discuss in Section 7.2 preconditioning methods such as substructuring, Schwarz overlapping, and multigrid techniques. Section 7.3 describes the mortar element method, which encompasses both functional and geometrical nonconforming discretizations; suggestions for implementation are given. Section 7.4 reviews the coupling between finite and spectral elements, provides some theoretical considerations about adaptivity near geometrical singularities, and considers for the 2D case hp-spectral triangular elements, which open the way for the coupling between quadrilaterals and triangles.

Preconditioning Methods

Domain-decomposition preconditioning has gained much attention over the past decade, both in theory and in practice.

According to the Greek philosopher Heraclitus, who used to say “πανταρ∈ι …,” daily life is concerned with the flow of ordinary fluids: water, air, blood, and so forth, in very common situations like breathing, coffee drinking, and hand washing.

Most flows are generated by nature (e.g., oceans, winds, rivers) and by human industrial activity (e.g., planes, cars, materials processing, biomedical engineering). There is a need to model fluid flow problems in order to improve the basic understanding of these complex phenomena and to increase the design quality of technological applications. With the advent of large and powerful computational tools, modeling has become more and more a substitute for direct experimentation. In some circumstances, experimentation may be too expensive – particularly if it leads to the destruction of the facility – or even impossible to perform, so that modeling is the only reasonable way to get answers and to study a range of parameters for optimal design.

Viscous Fluid Flows

We know from experience that many flows are set into motion by shear forces, and hence viscous effects play a vital role in fluids. In general, the viscosity depends on the shear rate (roughly speaking, the velocity gradient), as is explained by non-Newtonian theory. In this book, however, we will restrict ourselves mainly to the case of viscous Newtonian incompressible fluids in isothermal situations or under the influence of thermal convection as described by the Boussinesq approximation.

High-order methods have gained increasing attention in recent years. Their theoretical development has reached a high level of sophistication, and at the same time the range of applications has been broadening, including such diverse topics as global atmospheric modeling, aerodynamics, oceanography, thermal convection, and theoretical chemistry. Specialized conferences on the subject like the International Conference on Spectral Applications and High-Order Methods (ICOSAHOM) have been launched to bring mathematicians, engineers, and computer scientists together in order to stimulate further work in the field and to prospect new areas: high-order time schemes, treatment of singularities, complex geometries, mixed discretization techniques, domain decomposition, and parallelism. These topics were once considered as the stumbling block of spectral methods. As time goes on, this is no longer true, and high-order methods apply more and more to real-life engineering problems.

The monograph by Gottlieb and Orszag [163] and the book by Canuto et al. [64] remain milestones in the subject. They are cited in almost every paper written on the topic. Gottlieb and Orszag's monograph was the first on the subject and contains very little about applications. Moreover, it is silent on the topics mentioned above. Most of the developments covered by Canuto et al. are devoted to simple geometries, but a last chapter entitled “Domain Decomposition Methods” introduces extensions to more complex geometries. Recent achievements in the field of high-order methods have far-reaching consequences for geometrically complex configurations.