A general linear method approach to the design and optimization of efficient, accurate, and easily implemented time-stepping methods in CFD Victor DeCariaa,1,∗, Sigal Gottliebb, Zachary J. Granta,1, William J. Laytonc aOak Ridge National Laboratory, Computational and Applied Mathematics Group bMathematics Department, University of Massachusetts Dartmouth cDepartment of Mathematics, University of Pittsburgh Abstract In simulations of fluid motion time accuracy has proven to be elusive. We seek highly accurate methods with strong enough stability properties to deal with the richness of scales of many flows. These methods must also be easy to implement within current complex, possibly legacy codes. Herein we develop, analyze and test new time stepping methods addressing these two issues with the goal of accelerating the development of time accurate methods addressing the needs of applications. The new methods are created by introducing inexpensive pre-filtering and post-filtering steps to popular methods which have been implemented and tested within existing codes. We show that pre-filtering and post-filtering a multistep or multi-stage method results in new methods which have both multiple steps and stages: these are general linear methods (GLMs). We utilize the well studied properties of ∗Corresponding author Email addresses:
[email protected] (Victor DeCaria),
[email protected] (Sigal Gottlieb),
[email protected] (Zachary J. Grant),
[email protected] (William J. arXiv:2010.06360v1 [math.NA] 13 Oct 2020 Layton) 1This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05- 00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US govern- ment retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or repro- duce the published form of this manuscript, or allow others to do so, for US government purposes.