Difference Methods for Initial-Value Problems

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.