These techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition.
Numerical results illustrate the effectiveness of the procedures.
Thus, it is shown that some methods which are not algebraically stable, like the Lobatto IIIA or A-stable SIRK methods, are convergent for the class of stiff problems under consideration. Finally, some results on the existence and uniqueness of the Runge- Kutta solution are also presented. The long-time behaviour of numerical approximations to the solutions of a semilinear parabolic equation undergoing a Hopf bifurcation is studied in this paper. It is shown that the phase portrait of a supercritical Hopf bifurcation is correctly represented by Runge-Kutta time discretization. In particular, the bifurcation point and the Hopf orbits are approximated with higher order.