On the unique solvability of the Runge-Kutta equations

Summary We consider the existence of a unique solution to the systems of equations that arise when we apply a Runge-Kutta method to a stiff nonlinear system of differential equationsU=f(t, U), withf satisfying a one-sided Lipschitz condition with constant . For any given product h, whereh denotes the step size, we present algebraic conditions on the Runge-Kutta matrixA which are necessary and sufficient for unique solvability of the equations. As a second topic, we consider the question whether the solution to the system of equations is stable with respect to perturbations (known as BSI-stability). For this property also, necessary and sufficient algebraic conditions onA are presented.The research of this author has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.). This work was finished while this author was visiting the Massachusetts Institute of Technology with additional financial support provided by L.N. Trefethen from his U.S. National Science Foundation Presidential Young Investigator AwardPart of this research was done while this author was visiting the University of Leiden with an Erwin Schrödinger stipend from the Fonds zur Förderung der wissenschaftlichen Forschung and financial support of the Netherlands Organization for Scientific Research (N.W.O.)