The numerical solution of differential-algebraic systems by Runge-Kutta methods
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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Description of differentialalgebraic problems
RungeKutta methods for differentialalgebraic equations
Convergence for index 1 problems
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Algebraic Topology Analysis asymptotic expansion classical order coefficients component compute consider consistent initial values Const convergence results defined denote derivative differential-algebraic equations differential-algebraic systems Edited error estimate Euler method exact solution explicit Runge-Kutta methods exponentially decaying extrapolation Figure follows formula g(yn given gives global error Hairer half-explicit Implicit Function Theorem implicit Runge-Kutta implies index 2 problem index 2 system Inserting JACOBIAN Lemma linear Lobatto IIIC neighbourhood nonlinear system notation NSTEP numerical solution obtain order conditions order of convergence ordinary differential equations PARAMETERS Petzold power of h Proceedings proof of Theorem R(oo Radau IIA methods recursively Ring modulator Runge-Kutta matrix Runge-Kutta method applied Runge-Kutta methods Section Seminar simplified Newton method singular matrix step SUBROUTINE sufficiently small Taylor expansion Theorem 4.4 Theory Topology trees V(xn Vlll Wanner y-component yields yn+1 z(xn zn+1