Global error control in implicit parallel peer methods

Abstract

Recently, Schmitt, Weiner, and Erdmann have proposed an efficient family of numerical methods termed Implicit Parallel Peer (IPP) methods. They are a subclass of s -stage general linear methods of order s – 1. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff differential equations. The special construction of IPP methods allows for a parallel implementation, which is advantageous in modern high-performance computation environment. In this paper we add one more useful functionality to IPP methods, i.e. automatic global error control. We show that the global error estimation developed by Kulikov and Shindin in multistep formulas is suitable for the methods of Schmitt, Weiner and Erdmann. Moreover, that global error estimation can be done in parallel. An algorithm of efficient stepsize selection is also discussed here.