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

• Published in: Journal of computational physics 2022-04, Vol.455, p.110927, Article 110927
• Main Authors: DeCaria, Victor, Gottlieb, Sigal, Grant, Zachary J., Layton, William J.
• Format: Article
• Language: English
• Subjects: Accuracy; Complexity; Computational physics; Design optimization; Fluid filters; General linear methods; Methods; Navier-Stokes; Stability; Time discretization; Time filter
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110927

•We present new methods for time accurate simulations of fluids.•The new methods can be easily implemented in codes based on commonly used methods.•The new methods are optimized using General Linear Method theory. 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 GLMs to understand the accuracy and stability of filtered method, and to design optimal new filters for popular time-stepping methods. We present several new embedded families of high accuracy methods with low cognitive complexity and excellent stability properties. Numerical tests of the methods are presented, including ones finding failure points of some methods. Among the new methods presented is a novel pair of alternating filters for the Implicit Euler method which induces a third order, A-stable, error inhibiting scheme which is shown to be particularly effective.