In this paper implicit and explicit exact difference schemes (EDS) for system x ′ = Ax of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for problems with periodic solutions on very large time interval demonstrate the efficiency and exactness of the EDS compared with high-order numerical methods. This result can be extended for constructing EDS for general systems of n linear differential equations with constant coefficients and nonstandard finite difference (NSFD) schemes preserving stability properties for quasi-linear system of equations x ′ = Ax + f (x).

Abstract— The aim of this work is to construct high-order numerical schemes that preserve the dynamical properties of a mathematical model describing the spread of computer viruses on the Internet. For this purpose, we first apply Mickens’ methodology to formulate a dynamically consistent nonstandard finite difference (NSFD) scheme for the model under consid­eration. After that, the constructed NSFD scheme is combined with Richardson’s extrapolation method to generate higher-accuracy numerical approximations. The result is that we obtain extrapolated numerical schemes that not only preserve the dynamical proper­ties of the computer virus propagation model but also provide higher-accuracy numerical approximations. In addition, a set of numerical examples is conducted to illustrate and support the theoretical findings and to show the advantages of the proposed numerical schemes.