Discretizations of nonlinear differential equations using explicit nonstandard methods

In a recent paper, Chen and Solis investigated the appearance of spurious solutions when first-order ODEs are discretized using Runge–Kutta schemes. They concluded that the reliability of the numerical solutions to a particular ODE could be verified only by constructing several discrete models and comparing their numerical results with the known properties of the exact solutions. We demonstrate that by using nonstandard schemes, all the difficulties found by Chen and Solis can be eliminated, and that qualitatively correct numerical solutions are obtained for all values of the step size. We illustrate these issues by applying nonstandard finite-difference techniques to the logistic, sine, cubic, and Monod equations.