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Volume 40, Issue 1
Unconditionally Optimal Error Estimates of the Bilinear-Constant Scheme for Time-Dependent Navier-Stokes Equations

Huaijun Yang & Dongyang Shi

DOI: 10.4208/jcm.2007-m2020-0164

J. Comp. Math., 40 (2022), pp. 127-146.

Published online: 2021-11

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  • Abstract

In this paper, the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme. The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally, while the previous works require certain time-step restrictions. The analysis is based on an iterated time-discrete system, with which the error function is split into a temporal error and a spatial error. The $\tau$-independent ($\tau$ is the time stepsize) error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis, which implies that the numerical solution in $L^{\infty}$-norm is bounded. Thus optimal error estimates can be obtained in a traditional way. Numerical results are provided to confirm the theoretical analysis.

  • Keywords

Navier-Stokes equations, Unconditionally optimal error estimates, Bilinear-constant scheme, Time-discrete system.

  • AMS Subject Headings

65M60, 65M12, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

huaijunyang@zua.edu.cn (Huaijun Yang)

shi_dy@zzu.edu.cn (Dongyang Shi)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-127, author = {Yang , Huaijun and Shi , Dongyang}, title = {Unconditionally Optimal Error Estimates of the Bilinear-Constant Scheme for Time-Dependent Navier-Stokes Equations}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {40}, number = {1}, pages = {127--146}, abstract = {

In this paper, the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme. The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally, while the previous works require certain time-step restrictions. The analysis is based on an iterated time-discrete system, with which the error function is split into a temporal error and a spatial error. The $\tau$-independent ($\tau$ is the time stepsize) error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis, which implies that the numerical solution in $L^{\infty}$-norm is bounded. Thus optimal error estimates can be obtained in a traditional way. Numerical results are provided to confirm the theoretical analysis.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2007-m2020-0164}, url = {http://global-sci.org/intro/article_detail/jcm/19973.html} }
TY - JOUR T1 - Unconditionally Optimal Error Estimates of the Bilinear-Constant Scheme for Time-Dependent Navier-Stokes Equations AU - Yang , Huaijun AU - Shi , Dongyang JO - Journal of Computational Mathematics VL - 1 SP - 127 EP - 146 PY - 2021 DA - 2021/11 SN - 40 DO - http://doi.org/10.4208/jcm.2007-m2020-0164 UR - https://global-sci.org/intro/article_detail/jcm/19973.html KW - Navier-Stokes equations, Unconditionally optimal error estimates, Bilinear-constant scheme, Time-discrete system. AB -

In this paper, the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme. The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally, while the previous works require certain time-step restrictions. The analysis is based on an iterated time-discrete system, with which the error function is split into a temporal error and a spatial error. The $\tau$-independent ($\tau$ is the time stepsize) error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis, which implies that the numerical solution in $L^{\infty}$-norm is bounded. Thus optimal error estimates can be obtained in a traditional way. Numerical results are provided to confirm the theoretical analysis.

Yang , Huaijun and Shi , Dongyang. (2021). Unconditionally Optimal Error Estimates of the Bilinear-Constant Scheme for Time-Dependent Navier-Stokes Equations. Journal of Computational Mathematics. 40 (1). 127-146. doi:10.4208/jcm.2007-m2020-0164
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