Dissipativity of Runge-Kutta methods in Hilbert spaces

This paper concerns the discretization by Runge-Kutta methods of the initial value problem u(t) = f(u), under the dissipative structural condition that there exist alpha greater than or equal to 0, beta > 0, such that f : W --> H, Re[f(w), w](H) less than or equal to alpha - beta\w\(2)(H), For All w is an element of W, for complex Hilbert spaces W subset of or equal to H. It is shown that strong A-stability is necessary to ensure the dissipativity of the method, whilst algebraic stability plus \R(infinity)\ < 1 is a sufficient condition in the case of DJ-irreducible methods.