POSITIVITY OF RUNGE-KUTTA AND DIAGONALLY SPLIT RUNGE-KUTTA METHODS

In this paper we study positivity of general Runge-Kutta (RK) and diag onally 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 co mponents of the initial vector is preserved. First we state and prove a theorem that gives conditions under which a general RK or DSRK metho d is positive on arbitrary positive problem set. Then we study problem s 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 positivi ty, which is an inherent property of the scheme. This result ensures t hat we can construct DSRK methods which are unconditionally positive a nd 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 co ndition. In this way we can give a complete positivity analysis for RK methods. We calculate the positivity threshold for some methods, whic h are of practical interest. Finally we prove a theorem which generali zes the well-known result of Bolley and Crouzeix to nonlinear problems . (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.