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Solving Ordinary Differential Equations II

Stiff and Differential-Algebraic Problems

  • Book
  • © 1996

Overview

Authors:
  1. Ernst Hairer

    1. Section de Mathématiques, C.P. 240, Université de Genève, Genève 24, Switzerland

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  2. Gerhard Wanner

    1. Section de Mathématiques, C.P. 240, Université de Genève, Genève 24, Switzerland

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • The subject of this book is the solution of stiff differential equations and of differential-algebraic systems
  • This second completely revised and enlarged edition contains, in particular, such new material as: explicit method with optimal stability, new numerical tests, properties of error growth functions, recent progress in differential-algebraic equations and improved FORTRAN codes
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Series in Computational Mathematics (SSCM, volume 14)

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Table of contents (38 chapters)

  1. Front Matter

    Pages I-XV
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  2. Stiff Problems — One-Step Methods

    1. Front Matter

      Pages 1-1
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    2. Examples of Stiff Equations

      • Ernst Hairer, Gerhard Wanner
      Pages 2-14

    3. Stability Analysis for Explicit RK Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 15-39

    4. Stability Function of Implicit RK-Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 40-50

    5. Order Stars

      • Ernst Hairer, Gerhard Wanner
      Pages 51-70

    6. Construction of Implicit Runge-Kutta Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 71-90

    7. Diagonally Implicit RK Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 91-101

    8. Rosenbrock-Type Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 102-117

    9. Implementation of Implicit Runge-Kutta Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 118-130

    10. Extrapolation Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 131-142

    11. Numerical Experiments

      • Ernst Hairer, Gerhard Wanner
      Pages 143-166

    12. Contractivity for Linear Problems

      • Ernst Hairer, Gerhard Wanner
      Pages 167-179

    13. B-Stability and Contractivity

      • Ernst Hairer, Gerhard Wanner
      Pages 180-200

    14. Positive Quadrature Formulas and B-Stable RK-Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 201-214

    15. Existence and Uniqueness of IRK Solutions

      • Ernst Hairer, Gerhard Wanner
      Pages 215-224

    16. B-Convergence

      • Ernst Hairer, Gerhard Wanner
      Pages 225-238

  3. Multistep Methods for Stiff Problems

    1. Front Matter

      Pages 239-239
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    2. Stability of Multistep Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 240-249

    3. “Nearly” A-Stable Multistep Methods

      • Ernst Hairer, Gerhard Wanner
      Pages 250-260
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Keywords

About this book

"Whatever regrets may be, we have done our best." (Sir Ernest Shack­ 0 leton, turning back on 9 January 1909 at 88 23' South.) Brahms struggled for 20 years to write his first symphony. Compared to this, the 10 years we have been working on these two volumes may even appear short. This second volume treats stiff differential equations and differential algebraic equations. It contains three chapters: Chapter IV on one-step (Runge-Kutta) meth­ ods for stiff problems, Chapter V on multistep methods for stiff problems, and Chapter VI on singular perturbation and differential-algebraic equations. Each chapter is divided into sections. Usually the first sections of a chapter are of an introductory nature, explain numerical phenomena and exhibit numerical results. Investigations of a more theoretical nature are presented in the later sections of each chapter. As in Volume I, the formulas, theorems, tables and figures are numbered con­ secutively in each section and indicate, in addition, the section number. In cross references to other chapters the (latin) chapter number is put first. References to the bibliography are again by "author" plus "year" in parentheses. The bibliography again contains only those papers which are discussed in the text and is in no way meant to be complete.

Reviews

From the reviews of the second edition:

This is a superb book...Throughout, illuminating graphics, sketches and quotes from papers of researchers in the field add an element of easy informality and motivate the text." Mathematics Today

“This volume, on nonstiff equations, is the second of a two-volume set. This second volume treats stiff differential equations and differential-algebraic equations. … This book is highly recommended as a text for courses in numerical methods for ordinary differential equations and as a reference for the worker. It should be in every library, both academic and industrial.” (Teodora-Liliana Rădulescu, Zentralblatt MATH, Vol. 1192, 2010)

Authors and Affiliations

  • Section de Mathématiques, C.P. 240, Université de Genève, Genève 24, Switzerland

    Ernst Hairer, Gerhard Wanner

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