Comparing Multistep Methods Within Parametric Classes to Determine Viability in Solver Applications

Sammanfattning: In the 2017 paper by Arévalo and Söderlind, a framework was established for creating linear multistep methods of various classes based on a polynomial formulation which includes variable step size adaptivity. These classes are: k step explicit and implicit methods of order k and k+1 respectively for nonstiff problems (such as Adams methods), and k step implicit methods of order k for stiff problems (such as BDF methods). For each method class and order, all multistep methods of maximal order, including those which lack zero stability, are given by a parametrization depending on the method class and order. In this paper we conduct a pre-study on low order methods, comparing the properties of methods of the same class and order, and present experimental results when these methods are applied to simple test problems. We are motivated by the possibility of using method changes as a primary means of error control in solvers alongside traditional error control tools such as step size variability and order control. This paper also discusses some of the difficulties encountered during the research and concludes with questions for future study.