Behavior of Runge-Kutta discretizations near equilibria of index 2 differential algebraic systems

Authors:
Johannes Schropp
Affiliations:
Department of Mathematics and Statistics, University of Konstanz, P.O.Box 5560, D-78434 Konstanz, Germany
Venue:
Applied Numerical Mathematics
Year:
2002

Citing 4
Cited 1

Differential-algebraic equations index transformations

SIAM Journal on Scientific and Statistical Computing - Telecommunication Programs at U.S. Universities
Projected implicit Runge-Kutta methods for differential-algebraic equations

SIAM Journal on Numerical Analysis
Half-explicit Runge-Kutta methods for differential-algebraic systems of index 2

SIAM Journal on Numerical Analysis
Iterative solution of nonlinear equations in several variables

Iterative solution of nonlinear equations in several variables

Attracting sets in index 2 differential algebraic equations and in their Runge-Kutta discretizations

Nonlinear Analysis: Theory, Methods & Applications

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Abstract

We analyze Runge-Kutta discretizations applied to index 2 differential algebraic equations (DAE's) near equilibria. We compare the geometric properties of the numerical and the exact solutions. It is shown that projected and half-explicit Runge-Kutta methods reproduce the qualitative features of the continuous system in the vicinity of an equilibrium correctly. The proof combines cut-off and scaling techniques for index 2 differential algebraic equations with some invariant manifold results of Schropp [J. Schropp, Geometric properties of Runge-Kutta discretizations for index 2 differential algebraic equations, Konstanzer Schriften in Mathematik und Informatik 128] and classical results for discretized ordinary differential equations.