The second-order backward differentiation formula is unconditionally zero-stable

Abstract

Previous studies of the stability of the second-order backward differentiation formula have concluded that stability is possible only if restrictions are placed on the stepsize ratios, for example, limiting the ratio to some value less than 1 + 2 . However, actual implementations of the BDFs differ from the usual theoretical models of such methods; in particular, practical codes use scaled derivatives (e.g. EPISODE) or backward differences to represent current information about the solution. The representation makes no difference to truncation errors, but it has an important effect of the propagation of roundoff errors and of errors in the solution of the implicit equations. In this paper it is shown that the divided difference implementation of the variable coefficient (variable stepsize extension of the) second-order BDF is zero-stable for unrestricted stepsize ratios.