Scientific Computing with Ordinary Differential Equations

Folkmar Bornemann ist Professor für Numerische Mathematik und Wissenschaftliches Rechnen an der Technischen Universität München.

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in re search and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numeri cal and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs.

The textbook provides a sound fundamental introduction to the mathematical and numerical aspects of discretization methods for solving initial value problems in ordinary differential equations. It would be a useful and timely text for a graduate course in numerical analysis of ODEs. It is written at a level which is accessible to such an audience, covers a wide variety of topics, both classical and modern, and contains a generous supply of homework exercises at the end of each chapter.

1 Time-Dependent Processes in Science and Engineering.- 1.1 Newton's Celestial Mechanics.- 1.2 Classical Molecular Dynamics.- 1.3 Chemical Reaction Kinetics.- 1.4 Electrical Circuits.- Exercises.- 2 Existence and Uniqueness for Initial Value Problems.- 2.1 Global Existence and Uniqueness.- 2.2 Examples of Maximal Continuation.- 2.3 Structure of Nonunique Solutions.- 2.4 Weakly Singular Initial Value Problems.- 2.5 Singular Perturbation Problems.- 2.6 Quasilinear Differential-Algebraic Problems.- Exercises.- 3 Condition of Initial Value Problems.- 3.1 Sensitivity Under Perturbations.- 3.2 Stability of ODEs.- 3.3 Stability of Recursive Mappings.- Exercises.- 4 One-Step Methods for Nonstiff IVPs.- 4.1 Convergence Theory.- 4.2 Explicit Runge-Kutta Methods.- 4.3 Explicit Extrapolation Methods.- 5 Adaptive Control of One-Step Methods.- 5.1 Local Accuracy Control.- 5.2 Control-Theoretic Analysis.- 5.3 Error Estimation.- 5.4 Embedded Runge-Kutta Methods.- 5.5 Local Versus Achieved Accuracy.- Exercises.- 6 One-Step Methods for Stiff ODE and DAE IVPs.- 6.1 Inheritance of Asymptotic Stability.- 6.2 Implicit Runge-Kutta Methods.- 6.3 Collocation Methods.- 6.4 Linearly Implicit One-Step Methods.- Exercises.- 7 MultiStep Methods for ODE and DAE IVPs.- 7.1 Multistep Methods on Equidistant Meshes.- 7.2 Inheritance of Asymptotic Stability.- 7.3 Direct Construction of Efficient Multistep Methods.- 7.4 Adaptive Control of Order and Step Size.- Exercises.- 8 Boundary Value Problems for ODEs.- 8.1 Sensitivity for Two-Point EVPs.- 8.2 Initial Value Methods for Timelike EVPs.- 8.3 Cyclic Systems of Linear Equations.- 8.4 Global Discretization Methods for Spacelike EVPs.- 8.5 More General Types of BVPs.- 8.6 Variational Problems.- Exercises.- References.- Software.