The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods

Ernst Hairer, Michel Roche, Christian Lubich (auth.)9780387518602, 0-387-51860-6

The term differential-algebraic equation was coined to comprise differential equations with constraints (differential equations on manifolds) and singular implicit differential equations. Such problems arise in a variety of applications, e.g. constrained mechanical systems, fluid dynamics, chemical reaction kinetics, simulation of electrical networks, and control engineering. From a more theoretical viewpoint, the study of differential-algebraic problems gives insight into the behaviour of numerical methods for stiff ordinary differential equations. These lecture notes provide a self-contained and comprehensive treatment of the numerical solution of differential-algebraic systems using Runge-Kutta methods, and also extrapolation methods. Readers are expected to have a background in the numerical treatment of ordinary differential equations. The subject is treated in its various aspects ranging from the theory through the analysis to implementation and applications.

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Table of contents :
Description of differential-algebraic problems….Pages 1-13
Runge-Kutta methods for differential-algebraic equations….Pages 14-22
Convergence for index 1 problems….Pages 23-29
Convergence for index 2 problems….Pages 30-54
Order conditions of Runge-Kutta methods for index 2 systems….Pages 55-70
Convergence for index 3 problems….Pages 71-91
Solution of nonlinear systems by simplified Newton….Pages 92-98
Local error estimation….Pages 99-105
Examples of differential-algebraic systems and their solution….Pages 106-123