This paper deals with the numerical solution of initial value problems (IVPs), for systems of ordinary differential equations (ODEs), by an explicit fourth-order Runge-Kutta method (we will refer to it as the classical fourth-order method) with special nonlinear stability property indicated by the positivity. Stepsize conditions, guaranteeing this property based on general theory, have been studied earlier, see e.g. Hundsdorfer and Verwer (2003). In this paper we show that general obtained result on positivity for classical fourth-order method is somewhat too strict. We obtain new results for positivity which are important in practical applications. We provide some computational experiments to illustrate our results. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Two non-standard predictor-corrector type finite difference methods for a SIR epidemic model are proposed. The methods have useful and significant features, such as positivity, basic stability, boundedness and preservation of the conservation laws. The proposed schemes are compared with classical fourth order Runge–Kutta and non-standard difference methods (NSFD). The stability analysis is studied and numerical simulations are provided.
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