Filtered Implicit Second-Derivative Time-Stepping Methods for Stiff Initial Value Problems

Abstract This work focuses on the construction and numerical analysis of novel time-filtered second-derivative methods for stiff equations. The procedure is based on the recent first-derivative time filters (DeCaria et al., 2022 [6]). Such methods are developed by incorporating inexpensive pre-filtering and post-filtering steps into existing second-derivative schemes. We show that applying these filtering steps to second-derivative multistep or multi-stage methods results in new methods that combine features of both multistep and multi-stage approaches, referred to as second-derivative general linear methods (SGLMs). The well-established properties of SGLMs are utilized to analyze the accuracy and stability of the filtered methods and to design optimal new filters for time-stepping schemes. Several new embedded families of high-accuracy methods with low cognitive complexity and excellent stability characteristics are introduced. Finally, numerical experiments validate the stability and efficiency of the proposed methods.