Asymptotic stability of multistep methods for nonlinear delay differential equations

Abstract: This paper is concerned with the numerical solution of delay differential equations. The emphasis is on the nonlinear stability of multistep methods. It is shown that every A-stable linear multistep method with piecewise constant or linear interpolation can preserve the asymptotic stability of a class of nonlinear systems, which is an extension of the well known GP-stability result to the nonlinear case. As a comparison, it is also proved that strict stability at infinity is necessary to the asymptotic stability of one-leg methods for non-autonomous equations. [Copyright &y& Elsevier]