本书为全英文版，是一本备受推崇的有关偏微分方程数值技术的教科书，被国外多家知名大学指定为教材。包括牛津大学、马里兰大学、北卡罗来纳州立大学等。

本书讲解了求解偏微分方程的标准数值方法，也提供了该领域的最新技术。书中透彻地分析了各种方法的性质，严格地讨论了稳定性问题，提供了各种层次的例题和习题。全书结构清晰有序，叙述言简意赅，是数学、工程学及计算机科学专业学生学习偏微分方程数值解法首选入门教材。

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 0.method

  2.11 A maximum principle and convergence

  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

  Exercises

3   2.D and 3.D parabolic equations

  3.1  The explicit method in a rectilinear box

  3.2  An ADI method in two dimensions

  3.3  ADI and LOD methods in three dimensions

  3.4  Curved boundaries

  3.5  Application to general parabolic problems

  Bibliographic notes

  Exercises

4   Hyperbolic equations in one space dimension

  4.1  Characteristics

  4.2  The CFL condition

  4.3  Error analysis of the upwind scheme

  4.4  Fourier analysis of the upwind scheme

  4.5  The Lax.Wendroff scheme

  4.6  The Lax.Wendroff method for conservation laws

  4.7  Finite volume schemes

  4.8  The box scheme

  4.9  The leap.frog scheme

  4.10 Hamiltonian systems and symplectic integration schemes

  4.11 Comparison of phase and amplitude errors

  4.12 Boundary conditions and conservation properties

  4.13 Extensions to more space dimensions

  Bibliographic notes

  Exercises

5   Consistency, convergence and stability

  5.1  Definition of the problems considered

  5.2  The finite difference mesh and norms

  5.3  Finite difference approximations

  5.4  Consistency, order of accuracy and convergence

  5.5  Stability and the Lax Equivalence Theorem

  5.6  Calculating stability conditions

  5.7  Practical (strict or strong) stability

  5.8  Modified equation analysis

  5.9  Conservation laws and the energy method of analysis

  5.10 Summary of the theory

  Bibliographic notes

  Exercises   

6   Linear second order elliptic equations in

  two dimensions

  6.1  A model problem

  6.2  Error analysis of the model problem

  6.3  The general diffusion equation

  6.4  Boundary conditions on a curved boundary

  6.5  Error analysis using a maximum principle

  6.6  Asymptotic error estimates

  6.7  Variational formulation and the finite element method

  6.8  Convection.diffusion problems

  6.9  An example

  Bibliographic notes

  Exercises

7   Iterative solution of linear algebraic equations

  7.1  Basic iterative schemes in explicit form

  7.2  Matrix form of iteration methods and their convergence

  7.3  Fourier analysis of convergence

  7.4  Application to an example

  7.5  Extensions and related iterative methods

  7.6  The multigrid method

  7.7  The conjugate gradient method

  7.8  A numerical example: comparisons

  Bibliographic notes

  Exercises

References

Index