Toward a simple and accurate Lagrangian-based error estimator for the BDF algorithms and adaptive time-stepping
International Journal of Numerical Methods for Heat & Fluid Flow
ISSN: 0961-5539
Article publication date: 14 September 2023
Issue publication date: 22 November 2023
Abstract
Purpose
The purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer.
Design/methodology/approach
The Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms.
Findings
The validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations.
Originality/value
In contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer.
Keywords
Acknowledgements
This work was supported by The Science and Technology Project of China Three Gorges Corporation (Grant No. 202103404), Research on the International Standard for Compressed Air Energy Storage (Grant No. HNBZ21-I004) and The Mass Entrepreneurship and Innovation Project of State Grid Zhejiang Electric Power Co., Ltd. (Grant No. B711JZ220007).
Author contributions: Yazhou Wang and Dehong Luo contributed equally.
Citation
Wang, Y., Luo, D., Zhang, X., Wang, Z., Chen, H., Zhang, X., Xie, N., Mei, S., Xue, X., Zhang, T. and Tamma, K.K. (2023), "Toward a simple and accurate Lagrangian-based error estimator for the BDF algorithms and adaptive time-stepping", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 33 No. 12, pp. 4101-4127. https://doi.org/10.1108/HFF-03-2023-0161
Publisher
:Emerald Publishing Limited
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