On the positivity step size threshold of Runge-Kutta methods

However, for certain (stiff) IVPs with some particular initial vectors, e.g., for some “smooth” vectors in semi-discretized diffusion problems, we experience preservation of positivity with much larger step sizes than the strict positivity step size threshold. To catch this phenomenon, in the second part of the paper we construct positively invariant sets of positive vectors and derive step size thresholds for the discrete version of the positive invariance. The resulting threshold for discrete positive invariance is, roughly speaking, inverse proportional to the one-sided Lipschitz constant only and is shown in good accordance with some displayed computational experiments.