Positivity of an explicit Runge–Kutta method

This paper deals with the numerical solution of initial value problems (IVPs), for systems of ordinary differential equations (ODEs), by an explicit fourth-order Runge–Kutta method (we will refer to it as the classical fourth-order method) with special nonlinear stability property indicated by the positivity. Stepsize conditions, guaranteeing this property based on general theory, have been studied earlier, see e.g. Hundsdorfer and Verwer (2003). In this paper we show that general obtained result on positivity for classical fourth-order method is somewhat too strict. We obtain new results for positivity which are important in practical applications. We provide some computational experiments to illustrate our results.