An accurate and simple universal a posteriori error estimator for GS4-1 framework: Adaptive time stepping in first-order transient systems

Wang, Yazhou; Xue, Tao; Tamma, Kumar K.; Maxam, D.; Qin, Guoliang

doi:10.1016/j.cma.2020.113604

Cited by 13 publications

(8 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where ϕ = W 1 Λ 6 – W 1 Λ 4 . In this manner, the results of boldu˙n+1shift are second-order accurate (Wang et al , 2021c). Initial conditions : For a specific problem, the initial conditions are given for u ( t = 0) = u 0 in general.…”

Section: Numerical Methodologymentioning

confidence: 95%

“…With the adaptive time-stepping technique (see also Wang et al , 2021c; Wang et al , 2021d) and references therein), the time step is changed via the estimated local error and prescribed error threshold as demonstrated in Figure 2. It is worth noting that the relative error is most commonly used wherein the measure of error is divided by the corresponding measure of u n , that is: …”

Section: A-posteriori Error Estimator and Adaptive Time-steppingmentioning

confidence: 99%

“…3. A-posteriori error estimator and adaptive time-stepping A simple universal error estimator has been proposed in our previous work (Wang et al, 2021c) and applied for heat transfer problems. Herein in this section, we first review the theory part and then give a new and compact representation for simple numerical implementation, which will be used to solve the fluid dynamic problems in this paper.…”

Section: Application Of the Gs4-1 Frameworkmentioning

confidence: 99%

“…Regarding the LMS methods for first-order transient systems, the focus of this paper, a simple universal A -posteriori error estimator for the entire family of the GS4-1 framework was designed and analyzed with rigor via mathematical derivations and numerical demonstrations in the paper of Wang et al (2021c). A related work designed for the GS4-2 framework in second-order transient systems can be also found in the paper of Wang et al (2021d).…”

Section: Introductionmentioning

confidence: 99%

“…A noteworthy and useful interesting feature is that by merging the proposed error estimator to the adaptive time-stepping procedure, it is shown that the high-frequency oscillation existing in the Crank-Nicolson method can be effectively eliminated by the automatically selected time steps for fluid dynamic problems. More strictly, the high-frequency oscillations in damping systems like heat conduction problem (Wang et al, 2021c) and viscous Navier-Stokes equations could be suppressed by suitable time steps, but the adaptive time-stepping could not eliminate the high-frequency oscillation in energy-conserving structural dynamic second-order systems (see also Wang et al, 2021d), which is worth noting. In summary, this paper concentrates on the application of the GS4-1 framework for timedependent Navier-Stokes equations with automatic time step control strategy based on a proposed novel accurate error estimator in a compact representation, contributing to the efficient transient simulation in computational fluid dynamics.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Wang

Xie

Yin

et al. 2022

HFF

Self Cite

View full text Add to dashboard Cite

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.

show abstract