The Graduate Student's Guide to Numerical Analysis '98: Lecture Notes from the VIII Epsrc Summer School in Numerical Analysis (Hardcover, 1999)

A Simple Introduction to Error Estimation for Nonlinear Hyperbolic Conservation Laws.- 1 Introduction.- 2 Some Convection-Diffusion Problems.- 2.1 Traffic Flow.- 2.2 Propagation of Phase Transitions.- 2.3 Concluding Remarks.- 3 Continuous Dependence for Nonlinear Convection-Diffusion.- 3.1 The Standard Duality Technique and the Adjoint Problem.- 3.2 A Technique to Bypass the Resolution of the Adjoint Problem.- 3.3 A Very Simple Way of Handling the Convective Nonlinearity f.- 3.4 Continuous Dependence Results in L1-like Norms.- 3.5 Allowing the Diffusion Coefficients to Go to Zero.- 3.6 New Continuous Dependence Results.- 3.7 Relaxing the Smoothness in Time of the Approximate Solution u.- 3.8 The a Posteriori Error Estimate for Non-Smooth u.- 3.9 Concluding Remarks.- 4 Continuous Dependence for Nonlinear Convection.- 4.1 Existence and Uniqueness of the Entropy Solution.- 4.2 The Inherited Continuous Dependence Results.- 4.3 Concluding Remarks.- 5 A Posteriori Error Estimates for Continuous Approximations.- 5.1 The Error Estimate.- 5.2 Application to the Engquist-Osher Scheme.- 5.3 Explaining the Numerical Results.- 5.4 Another Error Estimate.- 6 A Posteriori Error Estimates for Discontinuous Approximations.- 6.1 The Case of a Finite Number of Smooth Discontinuity Curves.- 6.2 The Case of a Piecewise-Constant Approximation.- 7 Concluding Remarks.- 7.1 Some Bibliographical Remarks.- 7.2 Open Problems.- Notes on Accuracy and Stability of Algorithms in Numerical Linear Algebra.- 1 Introduction.- 2 Preliminaries.- 3 Symmetric Indefinite Systems.- 3.1 Block LDLT Factorization.- 3.2 Aasen's Method.- 3.3 Aasen's Method Versus Block LDLT Factorization.- 3.4 Tridiagonal Matrices.- 4 QR Factorization and Constrained Least Squares Problems.- 4.1 Householder QR Factorization.- 4.2 The Constrained Least Squares Problem.- 5 The Singular Value Decomposition and Jacobi's Method.- 5.1 Jacobi's Method.- 5.2 Relative Perturbation Theory.- 5.3 Error Analysis.- 5.4 Other Issues.- Numerical Analysis of Semilinear Parabolic Problems.- 1 The Continuous Problem.- 2 Local a Priori Error Estimates.- 2.1 The Spatially Semidiscrete Problem.- 2.2 A Completely Discrete Scheme.- 3 Shadowing-First Approach.- 3.1 Linearization.- 3.2 Exponential Dichotomies.- 3.3 Shadowing.- 4 A Posteriori Error Estimates.- 4.1 The Error Equation.- 4.2 Local Estimates of the Residual.- 4.3 A Global Error Estimate.- 5 Shadowing-Second Approach.- Integration Schemes for Molecular Dynamics and Related Applications.- 1 Introduction.- 2 Newtonian Dynamics.- 2.1 Properties.- 2.2 The Liouville Equation.- 3 The Leapfrog Method.- 3.1 Derivation.- 3.2 Small-?t Analysis.- 3.3 Linear Analysis.- 3.4 Small-Energy Analysis.- 3.5 Effective Accuracy and Post-Processing.- 3.6 Finite-Precision Effects.- 4 Other Methods.- 4.1 A Family of Methods.- 4.2 Quest for Accuracy and Stability.- 4.3 The Case for Symplectic Integration.- 5 Multiple Time Steps.- 5.1 The Verlet-I/r-RESPA/Impulse MTS Method.- 5.2 Partitioning of Interactions.- 5.3 Efficient Implementation.- 5.4 Mollified Impulse MTS Methods.- 6 Constrained Dynamics.- 6.1 Discretization.- 6.2 Solution of the Nonlinear Equations.- 7 Constant-Temperature and Constant-Pressure Ensembles.- 7.1 Constant-Temperature Ensembles.- 7.2 Constant-Pressure Ensembles.- 8 Stochastic Dynamics.- 8.1 Langevin Dynamics.- 8.2 Brownian Dynamics.- A Lie Series and the BCH Formula.- B Stochastic Processes.- 2.1 Wiener Processes.- 2.2 The Ito Integral.- 2.3 Stochastic Differential Equations.- 2.4 The Fokker-Planck Equation.- 2.5 The Ito Formula.- 2.6 Weak Ito-Taylor Expansions.- Numerical Methods for Bifurcation Problems.- 1 Introduction.- 2 Examples.- 3 Newton's Method and the Implicit Function Theorem.- 3.1 Newton's Method for Systems.- 3.2 The Implicit Function Theorem.- 3.3 Two Examples.- 4 Computation of Solution Paths.- 4.1 Keller's Pseudo-Arclength Continuation [25].- 4.2 Block Elimination.- 5 The Computation of Fold (Turning) Points.- 5.1 Analysis of Fold Poi