Positivity of Runge-Kutta and diagonally split Runge-Kutta methods

Abstract

In this paper we study positivity of general Runge-Kutta (RK) and diagonally split Runge-Kutta (DSRK) methods when applied to the numerical solution of positive initial value problems for ordinary differential equations. Here we mean by positivity that the nonnegativity of the components of the initial vector is preserved. First we state and prove a theorem that gives conditions under which a general RK or DSRK method is positive on arbitrary positive problem set. Then we study problems which are simultaneously positive and dissipative. For such problems we give the maximal step size that—under a solvability assumption on the algebraic equations defining the method—guarantees positivity. We show how the step size threshold is governed by the radius of positivity, which is an inherent property of the scheme. This result ensures that we can construct DSRK methods which are unconditionally positive and have an order higher than 1. Note that such a method does not exist between the classical methods. Investigating the radius of positivity of RK methods further we can get rid of the additional solvability condition. In this way we can give a complete positivity analysis for RK methods. We calculate the positivity threshold for some methods, which are of practical interest. Finally we prove a theorem which generalizes the well-known result of Bolley and Crouzeix to nonlinear problems.