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The Graduate Student's Guide to Numerical Analysis '98 : Lecture Notes from the VIII EPSRC Summer School in Numerical Analysis :  Lecture Notes from the VIII EPSRC Summer School in Numerical Analysis - Mark Ainsworth

The Graduate Student's Guide to Numerical Analysis '98 : Lecture Notes from the VIII EPSRC Summer School in Numerical Analysis

Lecture Notes from the VIII EPSRC Summer School in Numerical Analysis

By: Mark Ainsworth (Editor), Jeremy Levesley (Editor), Marco Marletta (Editor)

Hardcover Published: July 1999
ISBN: 9783540657521
Number Of Pages: 252

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This book contains detailed lecture notes on six topics at the forefront of current research in numerical analysis & applied mathematics. Each set of notes presents a self-contained guide to a current research area & has an extensive bibliography. In addition, most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, & proceed to current research topics. The reader should therefore be able to gain quickly an insight into the important results & techniques in each area without recourse to the large research literature. Current (unsolved) problems are also described & directions for future research are given. This book is also suitable for professional mathematicians who require a succint & accurate account of recent research in areas parallel to their own, & graduates in mathematical sciences.

Prefacep. V
A Simple Introduction to Error Estimation for Nonlinear Hyperbolic Conservation Lawsp. 1
Introductionp. 1
Some Convection-Diffusion Problemsp. 3
Traffic Flowp. 4
Propagation of Phase Transitionsp. 8
Concluding Remarksp. 10
Continuous Dependence for Nonlinear Convection-Diffusionp. 10
The Standard Duality Technique and the Adjoint Problemp. 11
A Technique to Bypass the Resolution of the Adjoint Problemp. 12
A Very Simple Way of Handling the Convective Nonlinearity fp. 14
Continuous Dependence Results in L1-like Normsp. 16
Allowing the Diffusion Coefficients to Go to Zerop. 18
New Continuous Dependence Resultsp. 21
Relaxing the Smoothness in Time of the Approximate Solution up. 24
The a Posteriori Error Estimate for Non-Smooth up. 27
Concluding Remarksp. 28
Continuous Dependence for Nonlinear Convectionp. 29
Existence and Uniqueness of the Entropy Solutionp. 29
The Inherited Continuous Dependence Resultsp. 30
Concluding Remarksp. 32
A Posteriori Error Estimates for Continuous Approximationsp. 32
The Error Estimatep. 32
Application to the Engquist-Osher Schemep. 33
Explaining the Numerical Resultsp. 34
Another Error Estimatep. 37
A Posteriori Error Estimates for Discontinuous Approximationsp. 39
The Case of a Finite Number of Smooth Discontinuity Curvesp. 39
The Case of a Piecewise-Constant Approximationp. 41
Concluding Remarksp. 43
Some Bibliographical Remarksp. 43
Open Problemsp. 43
Notes on Accuracy and Stability of Algorithms in Numerical Linear Algebrap. 47
Introductionp. 47
Preliminariesp. 47
Symmetric Indefinite Systemsp. 49
Block LDLT Factorizationp. 49
Aasen's Methodp. 54
Aasen's Method Versus Block LDLT Factorizationp. 59
Tridiagonal Matricesp. 59
QR Factorization and Constrained Least Squares Problemsp. 60
Householder QR Factorizationp. 61
The Constrained Least Squares Problemp. 67
The Singular Value Decomposition and Jacobi's Methodp. 70
Jacobi's Methodp. 71
Relative Perturbation Theoryp. 74
Error Analvsisp. 76
Other Issuesp. 78
Numerical Analysis of Semilinear Parabolic Problemsp. 83
The Continuous Problemp. 83
Local a Priori Error Estimatesp. 89
The Spatially Semidiscrete Problemp. 90
A Completely Discrete Schemep. 93
Shadowing-First Approachp. 94
Linearizationp. 95
Exponential Dichotomiesp. 98
Shadowingp. 101
A Posteriori Error Estimatesp. 105
The Error Equationp. 106
Local Estimates of the Residualp. 109
A Global Error Estimatep. 112
Shadowing-Second Approachp. 114
Integration Schemes for Molecular Dynamics and Related Applicationsp. 119
Introductionp. 119
Newtonian Dynamicsp. 121
Propertiesp. 121
The Liouville Equationp. 123
The Leapfrog Methodp. 125
Derivationp. 126
Small-¿t Analysisp. 128
Linear Analysisp. 130
Small-Energy Analysisp. 132
Effective Accuracy and Post-Processingp. 134
Finite-Precision Effectsp. 136
Other Methodsp. 137
A Family of Methodsp. 140
Quest for Accuracy and Stabilityp. 141
The Case for Symplectic Integrationp. 143
Multiple Time Stepsp. 145
The Verlet-I/r-RESPA/Impulse MTS Methodp. 146
Partitioning of Interactionsp. 149
Efficient Implementationp. 151
Mollified Impulse MTS Methodsp. 152
Constrained Dynamicsp. 153
Discretizationp. 154
Solution of the Nonlinear Equationsp. 156
Constant-Temperature and Constant-Pressure Ensemblesp. 156
Constant-Temperature Ensemblesp. 157
Constant-Pressure Ensemblesp. 159
Stochastic Dynamicsp. 159
Langevin Dynamicsp. 160
Brownian Dynamicsp. 161
Lie Series and the BCH Formulap. 162
Stochastic Processesp. 164
Wiener Processesp. 165
The Ito Integralp. 166
Stochastic Differential Equationsp. 167
The Fokker-Planck Equationp. 167
The Ito Formulap. 168
Weak Ito-Taylor Expansionsp. 169
Numerical Methods for Bifurcation Problemsp. 177
Introductionp. 177
Examplesp. 178
Newton's Method and the Implicit Function Theoremp. 183
Newton's Method for Systemsp. 183
The Implicit Function Theoremp. 184
Two Examplesp. 187
Computation of Solution Pathsp. 188
Keller's Pseudo-Arclength Continuation [25]p. 189
Block Eliminationp. 192
The Computation of Fold (Turning) Pointsp. 193
Analysis of Fold Pointsp. 194
Numerical Calculation of Fold Pointsp. 196
Bifurcation from the Trivial Solutionp. 197
Scalar Casep. 197
n-Dimensional Casep. 199
Bifurcation in Nonlinear ODEsp. 203
The Shooting Method for ODEsp. 204
Analysis of Parameter Dependent ODEsp. 207
Calculation of Fold Points in ODEs Using Shootingp. 208
Hopf Bifurcationp. 209
Calculation of a Hopf Bifurcation Pointp. 210
The Detection of Hopf Bifurcations in Large Systemsp. 212
Spectra and Pseudospectrap. 217
Eigenvaluesp. 217
Pseudospectrap. 225
A Matrix Examplep. 233
An Operator Examplep. 236
History of Pseudospectrap. 243
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540657521
ISBN-10: 3540657525
Series: SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 252
Published: July 1999
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 24.77 x 16.51  x 1.27
Weight (kg): 0.5