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。本书分为两部分，分别列入Springer《应用数学丛书》之22卷和33卷，内容取自作者在克罗拉多州立大学所开研究生课程的讲义。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、工程学学生学习的首选入门教材。

本书分为两卷(或称为两部分)，分别列入Springer《应用数学丛书》之22卷和33卷，内容取自作者在克罗拉多州立大学所开研究生课程的讲义。该课程讲授偏微分方程的差分方法，授课的对象为应用数学和工程专业的研究生。

本书的特点是强调实际上机操作，阅读本书需要具备一定的偏微分方程基础知识和编程能力。第1卷《偏微分方程的数值方法——有限差分法》(Numerical Partial Differential Equations:Finite Difference Methods)已购权影印出版，编号为WB3299。

Series Preface

Preface

Preface to Part 1

Contents of Part 1

8 Stability of Initial-Boundary-Value Schemes

 8.1 Introduction

 8.2 Stability

 8.2.1  Stability: An Easy Case

 8.2.2  Stability: Another Easy Case

 8.2.3  GKSO: General Theory

 8.2.4  Left Quarter Plane Problems

 8.3 Constructing Stable Difference Schemes

 8.4 Consistency and Convergence

 8.4.1  Norms and Consistency

 8.4.2  Consistency of Numerical Boundary Conditions

 8.4.3  Convergence Theorem: Gustafsson

 8.5 Schemes Without Numerical Boundary Conditions

 8.6 Parabolic Initial-Boundary-Value Problems

9 Conservation Laws

 9.1 Introduction

 9.2 Theory of Scalar Conservation Laws

 9.2.1  Shock Formation

 9.2.2  Weak Solutions

 9.2.3  Discontinuous Solutions

 9.2.4  The Entropy Condition

 9.2.5  Solution of Scalar Conservation Laws

 9.3 Theory of Systems of Conservation Laws

 9.3.1  Solutions of Riemann Problems

 9.4 Computational Interlude VI

 9.5 Numerical Solution of Conservation Laws

 9.5.1  Introduction

 9.6 Difference Schemes for Conservation Laws

 9.6.1  Consistency

 9.6.2  Conservative Schemes

 9.6.3  Discrete Conservation

 9.6.4  The Courant-Friedrichs-Lewy Condition

 9.6.5  Entropy

 9.7 Difference Schemes for Scalar Conservation Laws

 9.7.1  Definitions

 9.7.2  Theorems

 9.7.3  Godunov Scheme

 9.7.4  High Resolution Schemes

 9.7.5  Flux-Limiter Methods

 9.7.6  Slope-Limiter Methods

 9.7.7  Modified Flux Method

 9.8 Difference Schemes for K-System Conservation Laws

 9.9 Godunov Schemes

 9.9.1  Godunov Schemes for Linear K-System Conservation Laws

 9.9.2  Godunov Schemes for K-System Conservation Laws

 9.9.3  Approximate Riemann Solvers: Theory

 9.9.4  Approximate Riemann Solvers: Applications

 9.10 High Resolution Schemes for Linear K-System Conservation Laws

 9.10.1  Flux-Limiter Schemes for Linear K-System Conservation Laws

 9.10.2  Slope-Limiter Schemes for Linear K-System Conservation Laws

 9.10.3  A Modified Flux Scheme for Linear K-System Conservation Laws

 9.10.4  High Resolution Schemes for K-System Conservation Laws

 9.11 Implicit Schemes

 9.12 Difference Schemes for Two Dimensional Conservation Law

 9.12.1  Some Computational Examples

 9.12.2  Some Two Dimensional High Resolution Schemes

 9.12.3  The Zalesak-Smolarkiewicz Scheme

 9.12.4  A Z-S Scheme for Nonlinear Conservation Laws

 9.12.5  Two Dimensional K-System Conservation Laws

10 Elliptic Equations

 10.1 Introduction

 10.2 Solvability of Elliptic Difference Equations: Dirichlet Boundary Conditions

 10.3 Convergence of Elliptic Difference Schemes: Dirichlet Boundary Conditions

 10.4 Solution Schemes for Elliptic Difference Equations: Introduction

 10.5 Residual Correction Methods

 10.5.1  Analysis of Residual Correction Schemes

 10.5.2  Jacobi Relaxation Scheme

 10.5.3  Analysis of the Jacobi Relaxation Scheme

 10.5.4  Stopping Criteria

 10.5.5  Implementation of the Jacobi Scheme

 10.5.6  Gauss-Seidel Scheme

 10.5.7  Analysis of the Gauss-Seidel Relaxation Scheme

 10.5.8  Successive Overrelaxation Scheme

 10.5.9  Elementary Analysis of SOR Scheme

 10.5.10  More on the SOR Scheme

 10.5.11  Line Jacobi, Gauss-Seidel and SOR Schemes

 10.5.12  Approximating Wb: Reality

 10.6 Elliptic Difference Equations: Neumann Boundary Conditions

 10.6.1  First Order Approximation

 10.6.2  Second Order Approximation

 10.6.3  Second Order Approximation on an Offset Grid

 10.7 Numerical Solution of Neumann Problems

 10.7.1  Introduction

 10.7.2  Residual Correction Schemes

 10.7.3  Jacobi and Gauss-Seidel Iteration

 10.7.4  SOR Scheme

 10.7.5  Approximation of Wb

 10.7.6  Implementation: Neumann Problems

 10.8 Elliptic Difference Equatidns: Mixed Problems

 10.8.1  Introduction

 10.8.2  Mixed Problems: Solvability

 10.8.3  Mixed Problems: Implementation

 10.9 Elliptic Difference Equations: Polar Coordinates

 10.10 Multigrid

 10.10.1  Introduction

 10.10.2  Smoothers

 10.10.3  Grid Transfers

 10.10.4  Multigrid Algorithm

 10.11 Computational Interlude VII

 10.11.1  Blocking Out: Irregular Regions

 10.11.2  HW0.0.4

 10.12 ADI Schemes

 10.13 Conjugate Gradient Scheme

 10.13.1  Preconditioned Conjugate Gradient Scheme

 10.13.2  SSOR as a Preconditioner

 10.13.3  Implementation

 10.14 Using Iterative Methods to Solve Time Dependent Problems

 10.15 Using FFTs to Solve Elliptic Problems

 10.16 Computational Interlude VIII

11 Irregular Regions and Grids

 11.1 Introduction

 11.2 Irregular Geometries

 11.2.1  Blocking Out

 11.2.2  Map the Region

 11.2.3  Grid Generation

 11.3 Grid Refinement

   11.3.1  Grid Refinement: Explicit Schemes for Hyperbolic Problems

   11.3.2  Grid Refinement for Implicit Schemes

 11.4 Unstructured Grids

References

Index