The Graduate Student's Guide to Numerical Analysis '98

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.