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In simulations of fluid motion time accuracy has proven to be elusive. Weseek highly accurate methods with strong enough stability properties to dealwith the richness of scales of many flows. These methods must also be easy toimplement within current complex, possibly legacy codes. Herein we develop,analyze and test new time stepping methods addressing these two issues with thegoal of accelerating the development of time accurate methods addressing theneeds of applications. The new methods are created by introducing inexpensivepre-filtering and post-filtering steps to popular methods which have beenimplemented and tested within existing codes. We show that pre-filtering andpost-filtering a multistep or multi-stage method results in new methods whichhave both multiple steps and stages: these are general linear methods (GLMs).We utilize the well studied properties of GLMs to understand the accuracy andstability of filtered method, and to design optimal new filters for populartime-stepping methods. We present several new embedded families of highaccuracy methods with low cognitive complexity and excellent stabilityproperties. Numerical tests of the methods are presented, including onesfinding failure points of some methods. Among the new methods presented is anovel pair of alternating filters for the Implicit Euler method which induces athird order, A-stable, error inhibiting scheme which is shown to beparticularly effective.

A general linear method approach to the design and optimization of efficient, accurate, and easily implemented time-stepping methods in CFD