Marginal stability and stabilization in the numerical integration of ordinary differential equations

Abstract

Strongly stable and consistent multistep methods with maximum order are subject to marginal (or weak) stability. In this paper we introduce modified multistep methods whose coefficients depend linearly on the stepsize h h and a parameter L L in such a way that the order of the original method is not decreased. By choosing L L in a suitable manner (depending essentially on f y ( x , y ) {f_y}(x,y) of the differential equation y ′ = f ( x , y ) y’ = f(x,y) and on the growth parameters of the multistep method), marginal stability can be eliminated.