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**A strongly-consistent difference scheme for 3D nonlinear Navier-Stokes equations.**
*(English)*
Zbl 1483.65144

Summary: 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.

### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35Q30 | Navier-Stokes equations |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

68W30 | Symbolic computation and algebraic computation |

### Keywords:

difference algebra; difference Gröbner basis; finite difference scheme; Navier-Stokes equations; symbolic computation
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\textit{X. Zhang} and \textit{Y. Chen}, J. Syst. Sci. Complex. 34, No. 6, 2378--2395 (2021; Zbl 1483.65144)

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