Numerical positivity for Runge-Kutta methods

Abstract

Over the last years, a great effort has been done to develop Runge-Kutta methods preserving qualitative properties of the exact solution like monotonicity of positivity.Some results are available in the literature that ensure these properties under certain stepsize restictions. However, these results are given for the exact numerical solution whereas in practice the numerical solution available is only an approximation of it. For example, when implicit Runge-Kutta methods are used, the numerical solution obtained comes out from the inexact numerical resolution of nonlinear systems. The aim of this work is the study of the effective stepsize restrictions for positivity when implicit Runge-Kutta schemes are used. To achive this goal, we consider separately the problem of finding positive stage value predictors, and the analysis of the iterative scheme used, in this case Newton method.