Canonical fractional-step methods and consistent boundary conditions for the incompressible Navier-Stokes equations.

*(English)*Zbl 1074.76585Summary: An account of second-order fractional-step methods and boundary conditions for the incompressible Navier-Stokes equations is presented. The goals of the work were (i) identification and analysis of all possible splitting methods of second-order splitting accuracy, and (ii) determination of consistent boundary conditions that yield second-order-accurate solutions. Exact and approximate block-factorization techniques were used to construct second-order splitting methods. It has been found that only three canonical types (D, P, and M) of splitting methods are nondegenerate, and all other second-order splitting schemes are either degenerate or equivalent to them. Investigation of the properties of the canonical methods indicates that a method of type D is recommended for computations in which zero divergence is preferred, while a method of type P is better suited to cases where highly accurate pressure is more desirable. The consistent boundary conditions on the tentative velocity and pressure have been determined by a procedure that consists of approximation of the split equations and the boundary limit of the result. It has been found that the pressure boundary condition is independent of the type of fractional-step methods. The consistent boundary conditions on the tentative velocity were determined in terms of the natural boundary condition and derivatives of quantities available at the current time step (to be evaluated by extrapolation). Second-order fractional-step methods that admit the zero-pressure-gradient boundary condition have been derived by using a transformation that involves the ”delta form” pressure. The boundary condition on the new tentative velocity becomes greatly simplified due to improved accuracy built into the transformation.

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

##### Keywords:

second-order splitting schemes
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\textit{M. J. Lee} et al., J. Comput. Phys. 168, No. 1, 73--100 (2001; Zbl 1074.76585)

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##### References:

[1] | Armfield, S.; Street, R., The fractional step method for the navier – stokes equations on staggered grids: the accuracy of three variations, J. comput. phys., 153, 660, (1999) · Zbl 0965.76058 |

[2] | Beam, R.M.; Warming, R.F., An implicit factored scheme for the compressible navier – stokes equations, Aiaa j., 16, 393, (1978) · Zbl 0374.76025 |

[3] | Choi, H.; Moin, P., Effects of the computational time step on numerical solutions of turbulent flow, J. comput. phys., 113, 1, (1994) · Zbl 0807.76051 |

[4] | Chorin, A.J., Numerical solution of the navier – stokes equations, Math. comp., 22, 745, (1968) · Zbl 0198.50103 |

[5] | Dukowicz, J.K.; Dvinsky, A.S., Approximate factorization as a higher-order splitting for the implicit incompressible flow equations, J. comput. phys., 102, 336, (1992) · Zbl 0760.76059 |

[6] | Harlow, F.W.; Welch, J.E., Numerical calculation of time-dependent viscous incompressible flow of fluids with free surface, Phys. fluids, 8, 2182, (1965) · Zbl 1180.76043 |

[7] | Karniadakis, G.E.; Israeli, M.; Orszag, S.A., High-order splitting methods for the incompressible navier – stokes equations, J. comput. phys., 97, 414, (1991) · Zbl 0738.76050 |

[8] | Kim, J.; Moin, P., Application of a fractional-step method to incompressible navier – stokes equations, J. comput. phys., 59, 308, (1985) · Zbl 0582.76038 |

[9] | Orszag, S.A.; Israeli, M.; Deville, M.O., Boundary conditions for incompressible flows, J. sci. comput., 1, 75, (1986) · Zbl 0648.76023 |

[10] | Perot, J.B., An analysis of the fractional step method, J. comput. phys., 108, 51, (1993) · Zbl 0778.76064 |

[11] | Temam, R., Sur l’approximation de la solution des équations de navier – stokes par la méthode des pas fractionaries, I, Arch. ration. mech. anal., 33, 135, (1969) · Zbl 0195.46001 |

[12] | Temam, R., Sur l’approximation de la solution des équations de navier – stokes par la méthode des pas fractionaries, II, Arch. ration. mech. anal., 33, 377, (1969) · Zbl 0207.16904 |

[13] | Van Kan, J., A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. sci. stat. comput., 7, 870, (1986) · Zbl 0594.76023 |

[14] | R. F. Warming and R. M. Beam, On the construction and application of implicit factored schemes for conservation law, in Proceedings of the Symposium on Computational Fluid Dynamics, New York, April 16-17, 1977, SIAM-AMS Proc. (Soc. for Industr. & Appl. Math., Philadelphia, 1978), Vol. 11, pp. 85-129. |

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