This is the second edition of a book, first published in 1957, and is divided into two parts: the first contains the theory of difference methods and the second, applications of this theory.
Part 1 starts with a chapter in which the heat flow problem in its simplest form is treated. This is most appropriate because it makes it possible to introduce many of the important concepts and methods of the theory of difference approximations in a simple but still instructive way. Then finite-difference methods for general initial-value problems are introduced and the relations among stability, consistency and convergence are made clear.
The remaining chapters of Part 1 are devoted to specific classes of problems, beginning with constant coefficient problems and proceeding to more difficult cases: variable coefficient and nonlinear problems and mixed initial-boundary value problems. The most important results obtained after 1957 are included in this second edition. A merit of the book is that some of the proofs are simplifications of the original ones.
As already mentioned, the second part contains applications. Where the stability theory developed in Part 1 is inadequate (because it only concerns the behavior in the limit Δt→0), the book provides practical stability criteria.
All chapters of the book have been considerably revised since the first edition. There are three entirely new chapters: one concerning mixed initial-boundary value problems and one in Part 2 called Multi-dimensional-fluid dynamics.
The book is primarily written for users of difference methods, but it can be highly recommended to everyone interested in the subject.
Author(s): Robert D. Richtmyer, K. W. Morton
Edition: Second
Publisher: Interscience Publishers
Year: 1967
Language: English
Pages: 419
Part I
GENERAL CONSIDERATIONS
1 Introduction
1.1 Initial-Value Problems .
1.2 The Heat Flow Problem
1.3 Finite-Difference Equations
1.4 Stability .
1.5 Implicit Difference Equations .
1.6 The Truncation Error .
1.7 Rate of Convergence
1.8 Comments on High-Order Formulas and Rounding Errors
1.9 Outline of the Remainder of the Book
2 Linear Operators
2.1 The Function Space of an Initial- Value Problem
2.2 Banach Spaces
2.3 Linear Operators in a Banach Space .
2.4 The Extension Theorem
2.5 The Principle of Uniform Boundedness
2.6 A Fundamental Convergence Theorem
2.7 Closed Operators
3 Linear Difference Equations
3.1 Properly Posed Initial-Value Problems
3.2 Finite-Difference Approximations.
3.3 Con vergence .
3.4 Stability .
3.5 Lax's Equivalence Theorem
3.6 The Closed Operator A I
3.7 Inhomogeneous Problems .
3.8 Change of Norm.
3.9 Stability and Perturbations
4 Pure Initial-Value Problems with Constant Coefficients
4.1 The Class of Problems
4.2 Fourier Series and Integrals
Properly Posed Initial-Value Problems
The Finite-Difference Equations .
Order of Accuracy and the Consistency Condition
Stability .
The von Neumann Condition .
A Simple Sufficient Condition .
The Kreiss Matrix Theorem
The Buchanan Stability Criterion .
Further Sufficient Conditions for Stability
5 Linear Problems with Variable Coefficients; Non-Linear Problems
5.1 Introduction. 91
5.2 Alternative Definitions of Stability 9S
5.3 Parabolic Equations. 100
5.4 Dissipative Difference Schemes for Symmetric Hyperbolic Equations 108
5.5 Further Results for Symmetric Hyperbolic Equations 119
5.6 Non-Linear Equations with Smooth Solutions 124
6 Mixed Initial-Boundary- Value Problems
6.1 Introduction. 131
6.2 Basic] deas of the Energy Method. 132
6.3 Simple Examples of the Energy Method: Stable Choice of Approximations to Boundary Conditions and to Non-Linear Terms 137
6.4 Coupled Sound and Heat Flow 143
6.5 Mixed Problems for Symmetric Hyperbolic Systems 146
6.6 Normal Mode Analysis and the Godunov-Ryabenkii Stability Criterion 151
6.7 Application of the G-R Criterion to Mixed Problems 156
6.8 Conclusions 164
7 Multi-Level Difference Equations
7.1 Notation.
7.2 Auxiliary Banach Space
7.3 The Equivalence Theorem .
7.4 Consistency and Order of Accuracy
7.5 Example of Du Fort and Frankel.
7.6 Summary.
Part II
APPLICATIONS
Preface to Part II .
8 Diffusion and Heat Flow
8.1 Examples of Diffusion .
8.2 The Simplest Heat-Flow Problem.
8.3 Variable Coefficients.
8.4 Effect of Lower Order Terms on Stability
8.5 Solution of the Implicit Equations
8.6 A Non-Linear Problem.
8.7 Problems in Several Space Variables .
8.8 Alternating-Direction Methods.
8.9 Splitting and Fractional-Step Methods
9 The Transport Equation
9 .1 Physical Basis
9.2 The General Neutron Transport Equation
9.3 Homogeneous Slab: One Group
9.4 Homogeneous Sphere: One Group.
9.5 The it Spherical Harmonic" Method .
9.6 Slab: Difference System I for Hyperbolic Equations
9.7 A Paradox
9.8 Slab: Difference System II (Friedrichs)
9.9 Implicit Schemes.
9.10 The Wick-Chandrasekhar Method for the Slab
9.11 Equivalence of the Two Methods .
9.12 Boundary Conditions
9.13 Difference Systems I and II
9.14 System III: Forward and Backward Space Differences.
9.15 System IV (Implicit)
9.16 System V (Carlson's Scheme) .
9.17 Generalization of the Wick-Chandrasekhar Method.
9.18 The sn Method of Carlson (1953) .
9.19 A Direct Integration Method
10 Sound Waves
Physical Basis .
The Usual Finite-Difference Equation .
An Implicit System .
Coupled Sound and Heat Flow .
A Practical Stability Criterion .
11 Elastic Vibrations
11.1 Vibrations of a Thin Beam .
11.2 Explicit Difference Equations
An Implicit System .
Virtue of the Implicit System
Solution of Implicit Equations of Arbitrary Order
Vibration of a Bar Under Tension
12 Fluid Dynamics in One Space Variable
12.1 Introduction 288
12.2 The Eulerian Equations . 289
12.3 Difference Equations, Eulerian 290
12.4 The Lagrangean Equations . 293
12.5 Difference Equations, Lagrangean . 295
12.6 Treatment of Interfaces in the Lagrangean Formulation 298
12.7 Conservation-Law Form and the Lax-Wendroff Equations. 300
12.8 The Jump Conditions at a Shock 306
12.9 Shock Fitting . 308
12.10 Effect of Dissipation. 311
12.11 Finite-Difference Equations. 317
12.12 Stability of the Finite-Difference Equations . 320
12.13 Numerical Tests of the Pseudo-Viscosity Method. 324
12.14 The Lax-Wendroff Treatment of Shocks 330
12.15 The Method of S. K. Godunov . 338
12.16 Magneto-Fluid Dynamics 345
13 Multi-Dimensional Fluid Dynamics
13.1 Introduction
13.2 The Multi-Dimensional Fluid-Dynamic Equations
13.3 Properly and Improperly Posed Problems.
13.4 The Two-Step Lax-Wendroff or L-W Method.
13.5 The Viscosity Term for the L-W Method .
13.6 Piecewise Analytic Initial-Value Problems.
13.7 A Program for the Development of Methods .
13.8 Characteristics in Two-Dimensional Flow .
13.9 Shock Fitting in Two Dimensions .
13.10 The Problem of the Atmospheric Front
References
Subject Index.
