偏微分方程数值解（英文影印版）（第2版）

　　偏微分方程是构建科学、工程学和其他领域的数学模型的主要手段。一般情况下，这些模型都需要用数值方法去求解。本书提供了标准数值技术的简明介绍。借助抛物线型、双曲线型和椭圆型方程的一些简单例子介绍了常用的有限差分方法、有限元方法、有限体方法、修正方程分析、辛积分格式、对流扩散问题、多重网格、共轭梯度法。利用极大值原理、能量法和离散傅里叶分析清晰严格地处理了稳定性问题。本书全面讨论了这些方法的性质，并附有典型的图像结果，提供了不同难度的例子和练习。
　　 本书可作为数学、工程学及计算机科学专业本科教材，也可供工程技术人员和应用工作者参考。

1 introduction
2 parabolic equations in one space variable
2.1 introduction
2.2 a model problem
2.3 series approximation
2.4 an explicit scheme for the model problem
2.5 difference notation and truncation error
2.6 convergence of the explicit scheme
2.7 fourier analysis of the error
2.8 an implicit method
2.9 the thomas algorithm
2.10 the weighted average or o-method
2.11 a maximum principle and convergence for μ(1 - θ) ≤
2.12 a three-time-level scheme
2.13 more general boundary conditions
2.14 heat conservation properties
2.15 more general linear problems
2.16 polar co-ordinates
2.17 nonlinear problems
bibliographic notes

　　In the ten years since the first edition of this book was published, the numerical solution of PDEs has moved forward in many ways. But when we sought views on the main changes that should be made for this second edition, the general response was that we should not change the main thrust of the book or make very substantial changes. We therefore aimed to limit ourselves to adding no more than 10%-20% of new material and removing rather little of the original text: in the event, the book has increased by some 23%.
　　Finite difference methods remain the starting point for introducing most people to the solution of PDEs, both theoretically and as a tool for solving practical problems. So they still form the core of the book. But of course finite element methods, dominate the elliptic equation scene, and finite volume methods are the preferred approach to the approximation of many hyperbolic problems. Moreover, the latter formulation also forms a valuable bridge between the two main methodologies. Thus we have introduced a new section on this topic in Chapter 4; and this has also enabled us to reinterpret standard difference schemes such as the Lax-Wendroff method and the box scheme in this way, and hence for example show how they are simply extended to nonuniform meshes. In addition, the finite element section in Chapter 6 has been followed by a new section on convection-diffusion problems: this covers both finite difference and finite element schemes and leads to the introduction of Petrov-Galerkin methods.
　　The theoretical framework for finite difference methods has been well established now for some time and has needed little revision. However, over the last few years there has been greater interaction between methods to approximate ODEs and those for PDEs, and we have responded to this stimulus in several ways. Firstly, the growing interest in applying symplectic methods to Hamiltonian ODE systems, and extending the approach to PDEs, has led to our including a section on this topic in Chapter 4 and applying the ideas to the analysis of the staggered leapfrog scheme used to approximate the system wave equation. More generally, the revived interest in the method of lines approach has prompted a complete redraft of the section on the energy method of stability analysis in Chapter 5, with important improvements in overall coherence as well as in the analysis of particular cases. In that chapter, too, is a new section on modified equation analysis: this technique was introduced thirty years ago, but improved interpretations of the approach for such as the box scheme have encouraged a reassessment of its position; moreover, it is again the case that its use for ODE approximations has both led to a strengthening of its analysis and a wider appreciation of its importance.
　　Much greater changes to our field have occurred in the practical application of the methods we have described. And, as we continue to have as our aim that the methods presented should properly represent and introduce what is used in practice, we have tried to reflect these changes in this new edition. In particular, there has been a huge improvement in methods for the iterative solution of large systems of algebraic equations. This has led to a much greater use of implicit methods for time-dependent problems, the widespread replacement of direct methods by iterative methods in finite element modelling of elliptic problems, and a closer interaction between the methods used for the two problem types. The emphasis of Chapter 7 has therefore been changed and two major sections added. These introduce the key topics of multigrid methods and conjugate gradient methods, which have together been largely responsible for these changes in practical computations.
　　We gave serious consideration to the possibility of including a number of MATLAB programs implementing and illustrating some of the key methods. However, when we considered how very much more powerful both personal computers and their software have become over the last ten years, we realised that such material would soon be considered outmoded and have therefore left this aspect of the book unchanged. We have also dealt with references to the literature and bibliographic notes in the same way as in the earlier edition: however, we have collected both into the reference list at the end of the book.
　　Solutions to the exercises at the end of each chapter are available in the form of LATEX files. Those involved in teaching courses in this area can obtain copies, by email only, by applying to solutions@cambridge.org.
　　We are grateful to all those readers who have informed us of errors in the first edition. We hope we have corrected all of these and not introduced too many new ones. Once again we are grateful to our colleagues for reading and commenting on the new material.