A New Methodology for the Development of Efficient Multistep Methods for First–Order IVPs with Oscillating Solutions IV: The Case of the Backward Differentiation Formulae

Abstract

A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and amplification–factor derivatives. This paper will apply the theory for computing the phase–lag and amplification–factor, originally developed for implicit multistep methods, to a subset of implicit methods, called backward differentiation formulae (BDF), and will examine the impact of the phase–lag and amplification–factor derivatives on the efficiency of these strategies. Next, we will show you the stability zones of these brand-new approaches. Lastly, we will discuss the results of numerical experiments and draw some conclusions about the established approaches.

Keywords

• numerical solution
• initial value problems (IVPs)
• backward differentiation formulae (BDF)
• phase–fitting
• amplification–fitting
• multistep methods