A note on modified optimal linear multistep methods

Abstract

Modified optimal linear k k -step methods (whose coefficients depend on the stepsize and on a parameter L L ) are used for the numerical integration of systems of nonlinear ordinary differential equations. It is shown that, by choosing L L suitably (depending essentially on the growth parameters of the k k -step method and on the logarithmic norm of the Jacobian of the given system), weak stability does no longer occur, and one of two types of stability (called asymptotical relative and asymptotical absolute stability) may be obtained.