Stability of linear multistep methods on the imaginary axis

The stability of linear multistep methods of order higher than one is investigated for hyperbolic equations. By means of the Routh array and the Hermite-Biehler theorem, the stability boundary on the imaginary axis is expressed in terms of the error constant of the third order term. As a corollary we state the result that the stability boundary for methods of order higher than two, is at most 3, and this value is attained by the Milne-Simpson method.This work was done during the author's stay at the Mathematical Centre Amsterdam, and the University of Technology, Eindhoven.