Non-standard Runge-Kutta Schemes

In this paper, we improve the Runge-Kutta schemes by using non-standard frameworks for the numerical solution of first-order ODEs. By applying non-standard frameworks, the spurious solutions of Runge-Kutta schemes can be eliminated, and qualitatively correct numerical solutions will be derived for all values of the step size. The proposed schemes preserve some important features of the problem, e.g. fixed points and their stability, positivity and boundedness. The good performance of these schemes is illustrated for some ODEs.