A Class of Multirate Multiderivative Schemes

Abstract

In this work, we treat the numerical resolution of ordinary differential equations (ODEs) that contain both stiff and non-stiff terms, where these terms can be identified and separated, and where the stiff terms are 'easier' to evaluate than the non-stiff terms. In [Wensch, Knoth, Galant, BIT Numer Math 49 (2009), pp. 449-473], a class of so-called multirate schemes has been proposed to efficiently resolve said ODEs. Here, we extend this class of schemes by adding multiple temporal derivatives of the non-stiff part to the formulation. Order conditions and simplified order conditions of up to order four are derived in this work. Through this modification to the original algorithm, we can devise schemes with lesser stages at the same order. In particular, we devise a four-stage fourth-order scheme. The efficacy of the proposed methods is demonstrated through numerical experiments.