A note on multi-step difference schemes

Constructing multi-step difference schemes is important for solving ODE numerically. In this paper, by using an algebraic function approximation to the exponent function, linear multi-step schemes for solving ODE are deduced. The approximation order of this linear multi-step scheme equals that of the algebraic function approximation to ez. Moreover, we show that the linear nn-step difference scheme of order 2n2n is unstable, which is proved in a novel way.