ON ERROR GROWTH FUNCTIONS OF RUNGE-KUTTA METHODS

This paper studies estimates of the form //y(1) - (y) over cap(1)$// l ess than or equal to phi(h nu)//y(0) (y) over cap(0)$// where y(1), (y ) over cap(1)$ are the numerical solutions of a Runge-Kutta method app lied to a stiff differential equation satisfying a one-sided Lipschitz condition (with constant nu). An explicit formula for the optimal fun ction phi(x) is given, and it is shown to be superexponential, i.e., p hi(x(1))phi(x(2)) less than or equal to phi(x(1) + x(2)) if x(1) and x (2) have the same sign. As a consequence, results on asymptotic stabil ity are obtained. Furthermore, upper bounds for phi(x) are presented t hat can be easily computed from the coefficients of the method.