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A Strongly-Consistent Difference Scheme for 3D Nonlinear Navier-Stokes Equations

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Abstract

This paper constructs a strongly-consistent explicit finite difference scheme for 3D constant viscosity incompressible Navier-Stokes equations by using of symbolic algebraic computation. The difference scheme is space second order accurate and temporal first order accurate. It is proved that difference Gröbner basis algorithm is correct. By using of difference Gröbner basis computation method, an element in Gröbner basis of difference scheme for momentum equations is a difference scheme for pressure Poisson equation. The authors find that the truncation errors expressions of difference scheme is consistent with continuous errors functions about modified version of above difference equation. The authors prove that, for strongly consistent difference scheme, each element in the difference Gröbner basis of such difference scheme always approximates a differential equation which vanishes on the analytic solutions of Navier-Stokes equations. To prove the strongly-consistency of this difference scheme, the differential Thomas decomposition theorem for nonlinear differential equations and difference Gröbner basis theorems for difference equations are applied. Numerical test certifies that strongly-consistent difference scheme is effective.

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Acknowledgements

The authors are grateful to Professor Vladimir Gerdt for his help and advice related to the referenced theorem and the use of the maple packages DifferentialThomas.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Xiaojing Zhang & Yufu Chen

Authors
  1. Xiaojing Zhang
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  2. Yufu Chen
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Corresponding authors

Correspondence to Xiaojing Zhang or Yufu Chen.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant No. 11271363.

This paper was recommended for publication by Editor FENG Ruyong.

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Zhang, X., Chen, Y. A Strongly-Consistent Difference Scheme for 3D Nonlinear Navier-Stokes Equations. J Syst Sci Complex 34, 2378–2395 (2021). https://doi.org/10.1007/s11424-020-9325-3

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  • DOI: https://doi.org/10.1007/s11424-020-9325-3

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