Table Of ContentSpringer Series in 26
Computational
Mathematics
Editorial Board
R. Bank, La Jolla (CA)
R. L. Graham, La Jolla (CA)
J. Stoer, Wiirzburg
R. Varga, Kent (Ohio)
Springer-Verlag Berlin Heidelberg GmbH
Mark Ainsworth
Jeremy Levesley
Marco MarIetta (Eds.)
The Graduate
Student's Guide
to Numerical
Analysis '98
Lecture Notes from the VIII EPSRC
Summer School in Numerical Analysis
Springer
Editors:
Mark Ainsworth
Jeremy Levesley
Marco Marletta (e-mail: [email protected])
Department of Mathematics
and Computer Science
University of Leicester
University Road
LEI 7RH Leicester
United Kingdom
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
The graduate student's guide to numerical analysis '98: lecture notes from the VIII EPSRC
Summer School in Numerical Analysis I Mark Ainsworth ... (ed.). -Berlin; Heidelberg; New York;
Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1999
(Springer series in computational mathematics; 26)
Mathematics Subject Classification (1991): 15-XX, 34-XX, 35-XX, 65-XX,
70-XX,76-XX
ISSN 0179-3632
ISBN 978-3-642-08503-1 ISBN 978-3-662-03972-4 (eBook)
DOI 10.1007/978-3-662-03972-4
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Preface
The Eighth EPSRC Numerical Analysis Summer School was held at the Uni
versity of Leicester from the 5th to the 17th of July, 1998. This was the third
Numerical Analysis Summer School to be held in Leicester. The previous
meetings, in 1992 and 1994, had been carefully structured to ensure that
each week had a coherent 'theme'. For the 1998 meeting, in order to widen
the audience, we decided to relax this constraint. Speakers were chosen to
cover what may appear, at first sight, to be quite diverse areas of numeri
cal analysis. However, we were pleased with the extent to which the ideas
cohered, and particularly enjoyed the discussions which arose from differing
interpretations of those ideas. We would like to thank all six of our main
speakers for the care which they took in the preparation and delivery of their
lectures. In this volume we present their lecture notes in alphabetical rather
than chronological order. Nick Higham, Alastair Spence and Nick Trefethen
were the speakers in week 1, while Bernardo Cockburn, Stig Larsson and Bob
Skeel were the speakers in week 2.
Another new feature of this meeting compared to its predecessors was that
we had 'invited seminars'. A numer of established academics based in the UK
were asked to participate in the afternoon seminar program. These speak
ers were Phil Knight (Strathc1yde University), Karl Meerbergen (Rutherford
Appleton Laboratory) and Franc;oise Tisseur (Manchester University) during
the first week; John Barrett (Imperial College), Charlie Elliott (Sussex Uni
versity) and Petr Plechac (Rutherford Appleton Laboratory) in the second
week. We would like to thank them for supporting the Summer School by
presenting their research.
The other afternoon seminars were given by Ke Chen, Stuart Hawkins,
Carsten Keller, Mark Kiither, Elizabeth Larsson and Milan Mihailovic in
week 1, and Mark Arnold, Carsten Maple and Robert Scheichl in week 2.
When we announced the Summer School we also announced that in each week
we would give prizes for the best presentations by PhD students. These were
won by Marc Kiither and Elizabeth Larsson (week 1) and Robert Scheichl
(week 2), whom we warmly congratulate. Not all the people listed above were
eligible for the prize competition, but we shall not embarrass the 'oldies' by
listing them separately.
VI Preface
One tradition which we did not abandon was that of choosing two dis
tinguished UK academics to act as 'Local Experts'. We would like to thank
Jennifer Scott from Rutherford Appleton Laboratories and John Barrett from
Imperial College, Local Experts in weeks 1 and 2 respectively. They ensured
that the academic program ran smoothly and gave much time and thought
to the organization of the seminars and selection of invited seminar speakers.
They also found themselves in the difficult position of assisting us in judging
the student talks for the prizes. Gerald Moore from Imperial College was
also co-opted onto one of the prize adjudicating committees. We would like
to thank our Local Experts and Gerald Moore for successfully carrying out
the near impossible task of fair adjudication.
Finally, we thank the Engineering and Physical Sciences Research Council
for their financial support, which covered all the costs of the six main speakers
plus the accommodation costs of the UK participants.
Mark Ainsworth, Jeremy Levesley and Marco MarIetta
Leicester, November 1998.
Table of Contents
Preface........................................................ V
A Simple Introduction to Error Estimation for Nonlinear
Hyperbolic Conservation Laws
Bernardo Cockburn .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Introduction ................................................ . 1
2 Some Convection-Diffusion Problems .......................... . 3
2.1 Traffic Flow ............................................ . 4
2.2 Propagation of Phase Transitions .......................... 8
2.3 Concluding Remarks ..................................... 10
3 Continuous Dependence for Nonlinear Convection-Diffusion. . .. . .. 10
3.1 The Standard Duality Technique and the Adjoint Problem. . .. 11
3.2 A Technique to Bypass the Resolution of the Adjoint Problem. 12
3.3 A Very Simple Way of Handling the Convective Nonlinearity f 14
3.4 Continuous Dependence Results in Ll-like Norms. . . . . . . . . . .. 16
3.5 Allowing the Diffusion Coefficients to Go to Zero. . . . . . . . . . . .. 18
3.6 New Continuous Dependence Results.. . . .. . . .. . . .. . . . . . . . .. 21
3.7 Relaxing the Smoothness in Time of the Approximate Solution
u ...................................................... 24
3.8 The a Posteriori Error Estimate for Non-Smooth u . . .. . . . . . .. 27
3.9 Concluding Remarks. . .. . . .. . . . . . . .. . . .. . . .. . . .. . . . . . . . .. 28
4 Continuous Dependence for Nonlinear Convection. .. . . .. . . . . . . . .. 29
4.1 Existence and Uniqueness of the Entropy Solution ........... 29
4.2 The Inherited Continuous Dependence Results. . . . . . . . . . . . . .. 30
4.3 Concluding Remarks ..................................... 32
5 A Posteriori Error Estimates for Continuous Approximations. . . . .. 32
5.1 The Error Estimate. . . . . . . .. . . . . .. . . .. . . .. . . . . . . . . . . . . . .. 32
5.2 Application to the Engquist-Osher Scheme . . . . . . . . . . . . . . . . .. 33
5.3 Explaining the Numerical Results ... " . . .. . . .. . . . . . . . . . . . .. 34
5.4 Another Error Estimate .................................. 37
6 A Posteriori Error Estimates for Discontinuous Approximations. . .. 39
6.1 The Case of a Finite Number of Smooth Discontinuity Curves. 39
6.2 The Case of a Piecewise-Constant Approximation. . . . . . . . . . .. 41
VIII Table of Contents
7 Concluding Remarks ......................................... 43
7.1 Some Bibliographical Remarks.. .. . . . . .. . . . . .. . . . . .. . . . . . .. 43
7.2 Open Problems. . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . . .. 43
Notes on Accuracy and Stability of Algorithms in Numerical
Linear Algebra
Nicholas J. Higham ............................................. 47
1 Introduction................................................. 47
2 Preliminaries................................................ 47
3 Symmetric Indefinite Systems ................................. 49
3.1 Block LD LT Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
3.2 Aasen's Method ......................................... 54
3.3 Aasen's Method Versus Block LDLT Factorization. . . . .. . . . .. 59
3.4 Tridiagonal Matrices ..................................... 59
4 QR Factorization and Constrained Least Squares Problems. . . . . . .. 60
4.1 Householder QR Factorization.. . . . . .. . . . . . . . . . . . . . . .. . . . .. 61
4.2 The Constrained Least Squares Problem . . . . . . . . . . . . . . . . . . .. 67
5 The Singular Value Decomposition and Jacobi's Method. . . . . . . . .. 70
5.1 Jacobi's Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71
5.2 Relative Perturbation Theory... . . . . .. . . . . .. . . . . . . . . .. . . . .. 74
5.3 Error Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
5.4 Other Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78
Numerical Analysis of Semilinear Parabolic Problems
Stig Larsson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83
1 The Continuous Problem ..................................... 83
2 Local a Priori Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89
2.1 The Spatially Semi discrete Problem. . . .. . . . . . . .. . . . . . . .. . .. 90
2.2 A Completely Discrete Scheme. .. . . . . .. . . . . . . .. . . . . . . . . . .. 93
3 Shadowing-First Approach. . ... . .... . . . . .. . . . . .. . . . . . . . . . . . .. 94
3.1 Linearization............................................ 95
3.2 Exponential Dichotomies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98
3.3 Shadowing .............................................. 101
4 A Posteriori Error Estimates .................................. 105
4.1 The Error Equation ...................................... 106
4.2 Local Estimates of the Residual ............................ 109
4.3 A Global Error Estimate .................................. 112
5 Shadowing-Second Approach ................................. 114
Integration Schemes for Molecular Dynamics and Related
Applications
Robert D. Skeel ................................................. 119
1 Introduction ................................................. 119
Table of Contents IX
2 Newtonian Dynamics ......................................... 121
2.1 Properties ............................................... 121
2.2 The Liouville Equation ................................... 123
3 The Leapfrog Method ........................................ 125
3.1 Derivation .............................................. 126
3.2 Small-L1t Analysis ........................................ 128
3.3 Linear Analysis .......................................... 130
3.4 Small-Energy Analysis .................................... 132
3.5 Effective Accuracy and Post-Processing ..................... 134
3.6 Finite-Precision Effects ................................... 136
4 Other Methods .............................................. 137
4.1 A Family of Methods ..................................... 140
4.2 Quest for Accuracy and Stability ........................... 141
4.3 The Case for Symplectic Integration ........................ 143
5 Multiple Time Steps .......................................... 145
5.1 The Verlet-I/r-RESPA/Impulse MTS Method ............... 146
5.2 Partitioning of Interactions ................................ 149
5.3 Efficient Implementation .................................. 151
5.4 Mollified Impulse MTS Methods ........................... 152
6 Constrained Dynamics ........................................ 153
6.1 Discretization ........................................... 154
6.2 Solution of the Nonlinear Equations ........................ 156
7 Constant-Temperature and Constant-Pressure Ensembles ......... 156
7.1 Constant-Temperature Ensembles .......................... 157
7.2 Constant-Pressure Ensembles .............................. 159
8 Stochastic Dynamics ......................................... 159
8.1 Langevin Dynamics ...................................... 160
8.2 Brownian Dynamics ...................................... 161
A Lie Series and the BCH Formula ............................... 162
B Stochastic Processes .......................................... 164
2.1 Wiener Processes ........................................ 165
2.2 The Ito Integral ......................................... 166
2.3 Stochastic Differential Equations ........................... 167
2.4 The Fokker-Planck Equation .............................. 167
2.5 The Ito Formula ......................................... 168
2.6 Weak Ito-Taylor Expansions .............................. 169
Numerical Methods for Bifurcation Problems
Alastair Spence and Ivan G. Graham ............................. 177
1 Introduction................................................. 177
2 Examples ................................................... 178
3 Newton's Method and the Implicit Function Theorem ............ 183
3.1 Newton's Method for Systems ............................. 183
3.2 The Implicit Function Theorem ............................ 184
X Table of Contents
3.3 Two Examples ........................................... 187
4 Computation of Solution Paths ................................ 188
4.1 Keller's Pseudo-Arclength Continuation [25] ................ 189
4.2 Block Elimination ........................................ 192
5 The Computation of Fold (Turning) Points ...................... 193
5.1 Analysis of Fold Points ................................... 194
5.2 Numerical Calculation of Fold Points ....................... 196
6 Bifurcation from the Trivial Solution ........................... 197
6.1 Scalar Case ............................................. 197
6.2 n-Dimensional Case ...................................... 199
7 Bifurcation in Nonlinear ODEs ................................ 203
7.1 The Shooting Method for ODEs ........................... 204
7.2 Analysis of Parameter Dependent ODEs .................... 207
7.3 Calculation of Fold Points in ODEs Using Shooting .......... 208
8 Hopf Bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.1 Calculation of a Hopf Bifurcation Point ..................... 210
8.2 The Detection of Hopf Bifurcations in Large Systems ......... 212
Spectra and Pseudospectra
Lloyd N. Trefethen .............................................. 217
1 Eigenvalues ................................................. 217
2 Pseudospectra............................................... 225
3 A Matrix Example ........................................... 233
4 An Operator Example ........................................ 236
5 History of Pseudospectra ...................................... 243
