On error growth functions of Runge-Kutta methods Original Research Article

This paper studies estimates of the form View the MathML source, where y1, View the MathML source are the numerical solutions of a Runge-Kutta method applied to a stiff differential equation satisfying a one-sided Lipschitz condition (with constant ν). An explicit formula for the optimal function ϕ(x) is given, and it is shown to be superexponential, i.e., ϕ(x1)ϕ(x2) ≤ ϕ(x1 + x2) if x1 and x2 have the same sign. As a consequence, results on asymptotic stability are obtained. Furthermore, upper bounds for ϕ(x) are presented that can be easily computed from the coefficients of the method.