Scientific Computing with Ordinary Differential Equations

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.

P. Deuflhard and F. Bornemann

Scientific Computing with Ordinary Differential Equations

"Provides a sound fundamental introduction to the mathematical and numerical aspects of discretization methods for solving initial value problems in ordinary differential equations . . . This book would make an interesting (non-conventional) textbook 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. In summary, this is an excellent and timely book."MATHEMATICAL REVIEWS

"As indicated by the title, this is not a treatise merely on the numerical analysis of ordinary differential equations (ODEs). the reader is made acquainted with nontrivial examples of ODEs arising in diverse areas and with intrinsic properties of these equations. The consideration of all the various components makes for a very informative reading which is further aided by the undogmatic style of presentation. The volume can be recommended to newcomers, but also to instructors in this area." (H. Muthsam, Monatshefte für Mathematik, Vol. 143 (1), 2004)

Klappentext
This text provides an introduction to the numerical solution of initial and boundary value problems in ordinary differential equations on a firm theoretical basis. This book strictly presents numerical analysis as a part of the more general field of scientific computing. Important algorithmic concepts are explained down to questions of software implementation. For initial value problems, a dynamical systems approach is used to develop Runge-Kutta, extrapolation, and multistep methods. For boundary value problems including optimal control problems, both multiple shooting and collocation methods are worked out in detail.
Graduate students and researchers in mathematics, computer science, and engineering will find this book useful. Chapter summaries, detailed illustrations, and exercises are contained throughout the book with many interesting applications taken from a rich variety of areas.
Peter Deuflhard is founder and president of the Zuse Institute Berlin (ZIB) and full professor of scientific computing at the Free University of Berlin, Department of Mathematics and Computer Science.
Folkmar Bornemann is full professor of scientific computing at the Center of Mathematical Sciences, Technical University of Munich.
This book was translated by Werner Rheinboldt, professor emeritus of numerical analysis and scientific computing at the Department of Mathematics, University of Pittsburgh.

Inhalt
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.