2008 , Volume 13, № 3, p.54-64

A special one-step finite difference scheme for numerical analysis of the stiff Cauchy problem for a first order ordinary differential equation is proposed. It has uniform convergence on small parameter and is based on application of the asymptotic Laplace method for estimation of the integral solution of the Cauchy problem. A second order rational approximation of this special difference scheme is given. The comparison of the proposed difference scheme as well as its approximation efficiency with other known one-step difference methods is carried out on test problems.