Perturbed Runge--Kutta methods (also referred to as downwind Runge--Kutta
methods) can guarantee monotonicity preservation under larger step sizes
relative to their traditional Runge--Kutta counterparts. In this paper we
study, the question of how to optimally perturb a given method in order to
increase the radius of absolute monotonicity (a.m.). We prove that for methods
with zero radius of a.m., it is always possible to give a perturbation with
positive radius. We first study methods for linear problems and then methods
for nonlinear problems. In each case, we prove upper bounds on the radius of
a.m., and provide algorithms to compute optimal perturbations. We also provide
optimal perturbations for many known methods
