The Graduate Student's Guide to Numerical Analysis '98

by Epsrc Numerical Analysis Summer School 1998 (University of Leicester); Levesley, Jeremy; Marletta, Marco

• ISBN13: 9783540657521
• ISBN10: 3540657525
• Format: Hardcover
• Copyright: 1999-07-01
• Publisher: Springer Verlag

Summary

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

Table of Contents

Preface	v
A Simple Introduction to Error Estimation for Nonlinear Hyperbolic Conservation Laws	1	(46)
Bernardo Cockburn
Introduction	1	(2)
Some Convection-Diffusion Problems	3	(7)
Traffic Flow	4	(4)
Propagation of Phase Transitions	8	(2)
Concluding Remarks	10	(1)
Continuous Dependence for Nonlinear Convection-Diffusion	10	(19)
The Standard Duality Technique and the Adjoint Problem	11	(1)
A Technique to Bypass the Resolution of the Adjoint Problem	12	(2)
A Very Simple Way of Handling the Convective Nonlinearity f	14	(2)
Continuous Dependence Results in L1-like Norms	16	(2)
Allowing the Diffusion Coefficients to Go to Zero	18	(3)
New Continuous Dependence Results	21	(3)
Relaxing the Smoothness in Time of the Approximate Solution u	24	(3)
The a Posteriori Error Estimate for Non-Smooth u	27	(1)
Concluding Remarks	28	(1)
Continuous Dependence for Nonlinear Convection	29	(3)
Existence and Uniqueness of the Entropy Solution	29	(1)
The Inherited Continuous Dependence Results	30	(2)
Concluding Remarks	32	(1)
A Posteriori Error Estimates for Continuous Approximations	32	(7)
The Error Estimate	32	(1)
Application to the Engquist-Osher Scheme	33	(1)
Explaining the Numerical Results	34	(3)
Another Error Estimate	37	(2)
A Posteriori Error Estimates for Discontinuous Approximations	39	(4)
The Case of a Finite Number of Smooth Discontinuity Curves	39	(2)
The Case of a Piecewise-Constant Approximation	41	(2)
Concluding Remarks	43	(4)
Some Bibliographical Remarks	43	(1)
Open Problems	43	(4)
Notes on Accuracy and Stability of Algorithms in Numerical Linear Algebra	47	(36)
Nicholas J. Higham
Introduction	47	(1)
Preliminaries	47	(2)
Symmetric Indefinite Systems	49	(11)
Block LDLT Factorization	49	(5)
Aasen's Method	54	(5)
Aasen's Method Versus Block LDLT Factorization	59	(1)
Tridiagonal Matrices	59	(1)
QR Factorization and Constrained Least Squares Problems	60	(10)
Householder QR Factorization	61	(6)
The Constrained Least Squares Problem	67	(3)
The Singular Value Decomposition and Jacobi's Method	70	(13)
Jacobi's Method	71	(3)
Relative Perturbation Theory	74	(2)
Error Analysis	76	(2)
Other Issues	78	(5)
Numerical Analysis of Semilinear Parabolic Problems	83	(36)
Stig Larsson
The Continuous Problem	83	(6)
Local a Priori Error Estimates	89	(5)
The Spatially Semidiscrete Problem	90	(3)
A Completely Discrete Scheme	93	(1)
Shadowing---First Approach	94	(11)
Linearization	95	(3)
Exponential Dichotomies	98	(3)
Shadowing	101	(4)
A Posteriori Error Estimates	105	(9)
The Error Equation	106	(3)
Local Estimates of the Residual	109	(3)
A Global Error Estimate	112	(2)
Shadowing---Second Approach	114	(5)
Integration Schemes for Molecular Dynamics and Related Applications	119	(43)
Robert D. Skeel
Introduction	119	(2)
Newtonian Dynamics	121	(4)
Properties	121	(2)
The Liouville Equation	123	(2)
The Leapfrog Method	125	(12)
Derivation	126	(2)
Small-δt Analysis	128	(2)
Linear Analysis	130	(2)
Small-Energy Analysis	132	(2)
Effective Accuracy and Post-Processing	134	(2)
Finite-Precision Effects	136	(1)
Other Methods	137	(8)
A Family of Methods	140	(1)
Quest for Accuracy and Stability	141	(2)
The Case for Symplectic Integration	143	(2)
Multiple Time Steps	145	(8)
The Verlet-I/r-RESPA/Impulse MTS Method	146	(3)
Partitioning of Interactions	149	(2)
Efficient Implementation	151	(1)
Mollified Impulse MTS Methods	152	(1)
Constrained Dynamics	153	(3)
Discretization	154	(2)
Solution of the Nonlinear Equations	156	(1)
Constant-Temperature and Constant-Pressure Ensembles	156	(3)
Constant-Temperature Ensembles	157	(2)
Constant-Pressure Ensembles	159	(1)
Stochastic Dynamics	159	(3)
Langevin Dynamics	160	(1)
Brownian Dynamics	161	(1)
A Lie Series and the BCH Formula	162	(2)
B Stochastic Processes	164	(13)
Wiener Processes	165	(1)
The Ito Integral	166	(1)
Stochastic Differential Equations	167	(1)
The Fokker-Planck Equation	167	(1)
The Ito Formula	168	(1)
Weak Ito-Taylor Expansions	169	(8)
Numerical Methods for Bifurcation Problems	177	(40)
Alastair Spence
Ivan G. Graham
Introduction	177	(1)
Examples	178	(5)
Newton's Method and the Implicit Function Theorem	183	(5)
Newton's Method for Systems	183	(1)
The Implicit Function Theorem	184	(3)
Two Examples	187	(1)
Computation of Solution Paths	188	(5)
Keller's Pseudo-Arclength Continuation [25]	189	(3)
Block Elimination	192	(1)
The Computation of Fold (Turning) Points	193	(4)
Analysis of Fold Points	194	(2)
Numerical Calculation of Fold Points	196	(1)
Bifurcation from the Trivial Solution	197	(6)
Scalar Case	197	(2)
n-Dimensional Case	199	(4)
Bifurcation in Nonlinear ODEs	203	(6)
The Shooting Method for ODEs	204	(3)
Analysis of Parameter Dependent ODEs	207	(1)
Calculation of Fold Points in ODEs Using Shooting	208	(1)
Hopf Bifurcation	209	(8)
Calculation of a Hopf Bifurcation Point	210	(2)
The Detection of Hopf Bifurcations in Large Systems	212	(5)
Spectra and Pseudospectra	217
Lloyd N. Trefethen
Eigenvalues	217
Pseudospectra	225
A Matrix Example	233
An Operator Example	236
History of Pseudospectra	243