© 2001 by Institute of Mathematics and its Applications
Equilibrium attractivity of Runge–Kutta methods
1 Fachbereich Mathematik und Informatik, Universität Marburg, 35032 Marburg, Germany, e-mail: schmitt{at}mathmatik.uni-marburg.de 2 Fachbereich Mathematik und Informatik, Universität Halle, PF 8, 06099 Halle, Germany, e-mail: weiner{at}mathmatik.uni-marburg.de
For dissipative differential equations y' = f (y) it is known that contractivity of the exact solution is reproduced by algebraically stable Runge–Kutta methods. In this paper we investigate whether a different property of the exact solution also holds for Runge–Kutta solutions. This property, called equilibrium attractivity, means that the norm of the righthand side f never increases. It is a property dual to algebraic stability since neither is sufficient for the other, in general. We derive sufficient algebraic conditions for Runge–Kutta methods and prove equilibrium attractivity of the high-order algebraically stable Radau-IIA and Lobatto-IIIC methods and the Lobatto-IIIA collocation methods (which are not algebraically stable). No smoothness assumptions on f and no stepsize restrictions are required but, except for some simple cases, f has to satisfy certain additional properties which are generalizations of the simple one-sided Lipschitz condition using more than two argument points. These multipoint conditions are discussed in detail.
Key Words: stiff ODEs, implicit Runge–Kutta methods, equilibrium attractivity
Received 6 November 1998. Accepted 14 February 2000.