Reduction of computational errors using nonstandard finite difference models

Abstract

A previously reported bifurcation technique is applied to the construction of nonstandard finite difference representations of systems of nonlinear differential equations. This technique provides a measure of the deviation between bifurcation parameters obtained from fixed step representations of the nonlinear system and the values of the parameters determined from computational experiments. Since this deviation or ‘error’ is characteristic of a particular scheme, we have used this measure to construct low-error nonstandard representations. We present results from several nonlinear test models which show that such nonstandard schemes yield orbits that followed closely the expected dynamics and also provide a large reduction in the computational error in comparison to standard numerical integration schemes. Finally, we outline a criteria for controlling possible numerical overflow in fixed step-size schemes.