On the application of the GS4-1 framework for fluid dynamics and adaptive time-stepping via a universal A-posteriori error estimator

Yazhou Wang, Ningning Xie, Likun Yin, Tong Zhang, Xuelin Zhang, Shengwei Mei, Xiaodai Xue, Kumar Tamma

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Purpose: The purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational framework, applied for the fluid dynamics with illustrations to incompressible Navier–Stokes equations. Design/methodology/approach: The proposed error estimator is universal and versatile that it works for the entire subsets of the GS4-1 framework, encompassing the nondissipative Crank–Nicolson method, the most dissipative backward differential formula and anything in between. It is new and novel that the cumbersome design work of error estimation for specific time integration algorithms can be avoided. Regarding the numerical implementation, the local error estimation has a compact representation that it is determined by the time derivative variables at four successive time levels and only involves vector operations, which is simple for numerical implementation. Additionally, the adaptive time-stepping is further illustrated by the proposed error estimator and is used to solve the benchmark problems of lid-driven cavity and flow past a cylinder. Findings: The proposed computational procedure is capable of eliminating the nonphysical oscillations in GS4-1(1,1)/Crank–Nicolson method; being CPU-efficient in both dissipative and nondissipative schemes with better solution accuracy; and detecting the complex physics and hence selecting a suitable time step according to the user-defined error threshold. Originality/value: To the best of the authors’ knowledge, for the first time, this study applies the general purpose GS4-1 family of time integration algorithms for transient simulations of incompressible Navier–Stokes equations in fluid dynamics with constant and adaptive time steps via a novel and universal error estimator. The proposed computational framework is simple for numerical implementation and the time step selection based on the proposed error estimation is efficient, benefiting to the computational expense for transient simulations.

Original languageEnglish (US)
Pages (from-to)3306-3327
Number of pages22
JournalInternational Journal of Numerical Methods for Heat and Fluid Flow
Issue number10
StatePublished - Aug 16 2022

Bibliographical note

Funding Information:
This work is supported by The Science and Technology Project of China Three Gorges Corporation (Grant No. 202103404) and The Major Science and Technology Project of Inner Mongolia Autonomous Region (Grant No. 2021ZD0032). Acknowledgement is also due to Professor Tamma’s computational mechanics research lab at the University of Minnesota.

Publisher Copyright:
© 2022, Emerald Publishing Limited.


  • A-posteriori error estimator
  • Adaptive time-stepping
  • First-order transient systems
  • Fluid dynamics
  • Generalized single-step single-solve framework


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