Abstract :
In the first part of this paper we determine the largest step size of Runge–Kutta (RK) methods for which the corresponding numerical approximations are positive (component-wise non-negative) for arbitrary positive initial vector, whenever the underlying initial value problem (IVP) possesses the related positivity preserving property. We prove that step size thresholds for certain classes of positive IVPs guaranteeing positivity that we derived in a former paper are strict for irreducible and non-confluent RK methods. Investigating the strict positivity step size thresholds we can see that these are rather small if at all positive: often they are, roughly speaking, inverse proportional to the Lipschitz constant of the problem.
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.
