Systems of ordinary differential equations with solutions that are nonincreasing in an inner-product norm are considered in this paper. The technical assumptions imply that the systems are dissipative and no more than moderately stiff. Systems of this kind arise when parabolic partial differential equations are approximated by discretizing the spatial variables. Conditions are derived on Runge-Kutta methods and the step size, which guarantee that the numerical solution is nonincreasing. The correct qualitative behavior is certainly attractive, but it also allows convergence to be established on long time intervals.