A new class of highly-stable methods:A0-stable methods

AbstractA linear multistep method (ζ,σ) is defined to beA0-stable if when it is applied to the equation $$\dot x(t) = - \lambda x(t)$$ the approximate solutionxh(tn) tends to zero astn → ∞ for all values of the stepsizeh and allλ∈(0, ∞).Various properties ofA0-stable methods are derived. It is shown that most of the properties ofA(α)-stable methods are shared byA0-stable methods. It is proved that there existA0-stable methods of arbitrarily high order.