The Eighth EPSRC Numerical Analysis Summer School was held at the Uni­ versity of Leicester from the 5th to the 17th of July, 1998. 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.

A Simple Introduction to Error Estimation for Nonlinear Hyperbolic Conservation Laws.- 1 Introduction.- 2 Some Convection-Diffusion Problems.- 3 Continuous Dependence for Nonlinear Convection-Diffusion.- 4 Continuous Dependence for Nonlinear Convection.- 5 A Posteriori Error Estimates for Continuous Approximations.- 6 A Posteriori Error Estimates for Discontinuous Approximations.- 7 Concluding Remarks.- Notes on Accuracy and Stability of Algorithms in Numerical Linear Algebra.- 1 Introduction.- 2 Preliminaries.- 3 Symmetric Indefinite Systems.- 4 QR Factorization and Constrained Least Squares Problems.- 5 The Singular Value Decomposition and Jacobi’s Method.- Numerical Analysis of Semilinear Parabolic Problems.- 1 The Continuous Problem.- 2 Local a Priori Error Estimates.- 3 Shadowing—First Approach.- 4 A Posteriori Error Estimates.- 5 Shadowing—Second Approach.- Integration Schemes for Molecular Dynamics and Related Applications.- 1 Introduction.- 2 Newtonian Dynamics.- 3 The Leapfrog Method.- 4 Other Methods.- 5 Multiple Time Steps.- 6 Constrained Dynamics.- 7 Constant-Temperature and Constant-Pressure Ensembles.- 8 Stochastic Dynamics.- Numerical Methods for Bifurcation Problems.- 1 Introduction.- 2 Examples.- 3 Newton’s Method and the Implicit Function Theorem.- 4 Computation of Solution Paths.- 5 The Computation of Fold (Turning) Points.- 6 Bifurcation from the Trivial Solution.- 7 Bifurcation in Nonlinear ODEs.- 8 Hopf Bifurcation.- Spectra and Pseudospectra.- 1 Eigenvalues.- 2 Pseudospectra.- 3 A Matrix Example.- 4 An Operator Example.- 5 History of Pseudospectra.

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. Recent graduates in mathematical sciences who are considering the possibility of a pursuing research towards a PhD in applied mathematics or numerical analysis should also find this book a very valuable guide.

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