List of Contributors ............................................ v
Preface ....................................................... vii
A Finite Difference Scheme for Generalized Neumann Problems
K.O. Friedrichs and H.B. Keller
1 Introduction ................................................. 1
2 The Difference Method and Convergence for the Neumann
Problem ...................................................... 2
3 The Finite Difference Equations .............................. 7
4 Interpretation of the Finite Difference Equations ........... 10
5 Systems with Variable Coefficients .......................... 13
References ..................................................... 19
Remarks on the Order of Convergence in the Discrete Dirichlet
Problem
Bert Hubbard
1 Introduction ................................................ 21
2 Definitions ................................................. 22
3 The Dirichlet Problem ....................................... 24
4 Estimates for Derivatives (Richardson's Method) ............. 28
5 Problems with Singularities (Exterior Problems) ............. 31
References ..................................................... 34
Fluid Dynamical Calculations
Eugene Isaacson
1 Introduction ................................................ 35
2 Motion of Cold Fronts ....................................... 35
3 Flood Waves in Rivers ....................................... 42
4 Summary and Acknowledgments ................................. 48
References ..................................................... 48
Difference Approximations for Hyperbolic Differential
Equations
Heinz-Otto Kreiss
1 Cauchy Problem .............................................. 51
2 Initial-Boundary Value Problem .............................. 54
References ..................................................... 57
Discrete Methods for Nonlinear Two-Point Boundary Value
Problems
Milton Lees
1 Introduction ................................................ 59
2 A Discrete Boundary Value Problem ........................... 63
3 Newton's Method ............................................. 67
4 The Difference Correction ................................... 69
References ..................................................... 72
Remarks on the Numerical Computation of Solutions
of Δ = ƒ(P, u)
Seymour V. Parter
1 Introduction ................................................ 73
2 Analytic Preliminaries ...................................... 73
3 Finite-Difference Equations ................................. 76
4 Convergence Theorems ........................................ 79
References ..................................................... 82
Error Bounds Based on A Priori Inequalities
Lawrence E. Payne
1 Introduction ................................................ 83
2 A Priori Bounds ............................................. 83
References ..................................................... 92
On Admissibility in Representations of Functions of Several
Variables as Finite Sums of Functions of One Variable
David A. Sprecher
1 Introduction and Summary .................................... 95
2 On Separation of Points ..................................... 98
3 A Partition of Functions ................................... 103
4 The Problem of Admissibility ............................... 105
References .................................................... 108
Stability of Nonlinear Discretization Algorithms
Hans J. Stetter
Text .......................................................... 111
References .................................................... 123
On Maximum-Norm Stable Difference Operators
Vidar Thomee
Text .......................................................... 125
References .................................................... 151
A Posteriori Error Bounds in Iterative Matrix Inversion
H.F. Weinberger
1 Introduction ............................................... 153
2 The Symmetric Case ......................................... 156
3 Some Numerical Results ..................................... 159
4 An Improved Approximation .................................. 161
5 Remarks on Nonsymmetric Matrices ........................... 163
References .................................................... 163
Finite Difference Methods for Solving Systems of Partial
Differential Equations
J. Flügge-Lotz
Text .......................................................... 165
Some Numerical Results in Intermediate Problems for
Eigenvalues
A. Weinstein
Text .......................................................... 167
References .................................................... 189
Stability of Linear and Nonlinear Difference Schemes
Peter D. Lax
Text .......................................................... 193
References .................................................... 195
The Solutions of Multidimensional Generalized Transport
Equations and Their Calculation by Difference Methods
Avron Douglis
Introduction .................................................. 197
1 Statement of Problem; Notation; Minimal Assumptions ........ 197
2 Reduction of Problem to an Integral Equation ............... 202
3 Some Appropriate Function Spaces ........................... 208
4 Continuous Dependence, Uniqueness, and Existence ........... 210
5 A Priori Lipschitz Conditions for Weak Solutions;
Lipschitz Condition with Respect to г ...................... 216
6 An A Priori Lipschitz Condition for и with Respect to
the x8, s = 1, ..., d ...................................... 221
7 Determination and Behavior of t* and x* .................... 224
8 Proofs of Auxiliary Estimates .............................. 229
9 Positivity and Monotonic Dependence ........................ 237
10 Truncation ................................................. 239
11 Difference Scheme Notation; Some Remarks ................... 242
12 Statement of Difference Equations .......................... 246
13 Outline of Convergence Proof ............................... 249
14 A Bound for the Solution of the Difference Equations and
an Estimate for Its t Difference Quotients ................. 250
15 Boundary Behavior .......................................... 252
16 Lipschitz Conditions in Truncated Problems ................. 253
References .................................................... 256
Application of Integral Operators to Singular Differential
Equations and to Computations of Compressible Fluid Flows
Stefan Bergman
1 Introduction ............................................... 257
2 The Integral Operator Generating Solutions of Eq. (1.1) .... 259
3 The Equation for the Stream Function ф of Compressible
Fluid in the Pseudo-logarithmic Plane ...................... 268
4 A Set Ψν,ν = 1, 2, . . . , of Particular Solutions of Eq.
(3.3) Obtained Using the Generating Function (3.13) ........ 276
5 Remarks on Application of Integral Operators for
Numerical Purposes ......................................... 280
References .................................................... 283
Numerical Solution of the Telegraph and Related Equations
Garrett Birkhoff and Robert E. Lynch
A Introduction
1 Telegraph Equation ......................................... 289
2 Infinitesimal Amplification Factors ........................ 291
3 Relevant Difference Schemes ................................ 293
B Thermal Transients in Reactor Channels
4 Physical Problem ........................................... 295
5 Simplified Model ........................................... 296
6 Conduction Equation ........................................ 297
7 Convection Equation ........................................ 298
C Regenerator Problems
8 The Mathematical Model ..................................... 301
9 Asymptotic Discussion ...................................... 303
10 Moments .................................................... 304
11 Characteristic Triangular Meshes ........................... 308
D Generalizations
12 Hyperbolic Systems of Positive Type ........................ 309
13 Asymptotic Behavior ........................................ 312
References .................................................... 314
Approximation and Estimates for Eigenvalues
Gaetano Fichera
1 Eigenvalue Problems, the Rayleigh-Ritz Method .............. 318
2 The Weinstein-Aronszajn Method ............................. 323
3 Construction of the Intermediate Operators ................. 328
4 Orthogonal Invariants of Positive Compact Operators ........ 332
5 Upper Approximation of the Eigenvalues of a PCO;
Representation of Orthogonal Invariants .................... 338
Bibliography .................................................. 351
Approximate Continuation of Harmonic and Parabolic Functions
Jim Douglas, Jr.
1 Introduction ............................................... 353
2 Harmonic Continuation in a Disk ............................ 353
3 Harmonic Continuation in a Half-Plane ...................... 360
4 Continuation of Solutions of the Heat Equation ............. 361
References .................................................... 363
Hermite Interpolation-Type Ritz Methods for Two-Point
Boundary Value Problems
Richard S. Varga
1 Introduction ............................................... 365
2 The Spaces H(m)N and H ...................................... 365
3 Accuracy ................................................... 367
4 Determination of the Best Approximation in H(m)N ............ 369
5 Connection with Other Methods .............................. 370
6 A Posteriori Error Bounds and Spines ....................... 372
References .................................................... 373
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