Fast preconditioned multigrid solution of the Euler and Navier-Stokes equations for steady, compressible flows.

*(English)*Zbl 1032.76611Summary: New versions of implicit algorithms are developed for the efficient solution of the Euler and Navier-Stokes equations of compressible flow. The methods are based on a preconditioned, lower-upper (LU) implementation of a non-linear, symmetric Gauss-Seidel (SGS) algorithm for use as a smoothing algorithm in a multigrid method. Previously, this method had been implemented for flows in quasi-one-dimensional ducts and for two-dimensional flows past airfoils on boundary-conforming ‘O’-type grids for a variety of symmetric limited positive (SLIP) spatial approximations, including the scalar dissipation and convective upwind split pressure (CUSP) schemes. Here results are presented for both inviscid and viscous (laminar) flows past airfoils on boundary-conforming ‘C’-type grids. The method is significantly faster than earlier explicit or implicit methods for inviscid problems, allowing solution of these problems to the level of truncation error in three to five multigrid cycles. Viscous solutions still require as many as twenty multigrid cycles.

##### Keywords:

Euler equations; implicit methods; lower-upper schemes; multigrid methods; Navier-Stokes equations; symmetric Gauss-Seidel iteration; transonic flow
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\textit{D. A. Caughey} and \textit{A. Jameson}, Int. J. Numer. Methods Fluids 43, No. 5, 537--553 (2003; Zbl 1032.76611)

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##### References:

[1] | Jameson, Applied Mathematics and Computation 13 pp 327– (1983) |

[2] | Numerical solutions of the Euler equations by finite-volume methods using Runge-Kutta time-stepping schemes. AIAA Paper 81-1259, 1981. |

[3] | Pierce, Journal of Computational Physics 136 pp 425– (1997) |

[4] | Caughey, AIAA Journal 26 pp 841– (1988) |

[5] | Implicit multigrid Euler solutions with symmetric total variation diminishing dissipation. Proceedings of AIAA 11th CFD Conference, 1993; 676-684. |

[6] | Jameson, AIAA Journal 24 pp 1737– (1986) |

[7] | Solution of steady three-dimensional compressible Euler and Navier-Stokes equations by an implicit LU scheme. AIAA Paper 88-0619, 1988. |

[8] | Yoon, AIAA Journal 26 pp 1025– (1988) |

[9] | Yokota, AIAA Journal 26 pp 1061– (1988) |

[10] | Brandt, Mathematics of Computation 31 pp 333– (1977) |

[11] | Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics. GMD-Studie, vol.85. GMD-FIT, 1985. |

[12] | Textbook multigrid efficiency for the steady Euler equations. AIAA Paper 97-1949, 1997. |

[13] | Multigrid relaxation of a factorizable, conservative discretization of the compressible flow equations. AIAA Paper 2000-2252, 2000. |

[14] | Canonical-variables multigrid method for steady-state Euler equations. ICASE Report 94-14, 1994. |

[15] | Beam, Journal of Computational Physics 22 pp 87– (1976) |

[16] | Beam, AIAA Journal 16 pp 393– (1978) |

[17] | Solution of the Three-Dimensional Compressible Navier Stokes Equations by an Implicit Technique. Lecture Notes in Physics, vol.35. Springer: New York, 1974; 105-110. |

[18] | Jameson, Mathematics of Computation 37 pp 385– (1981) |

[19] | Buratynski, AIAA Journal 24 pp 39– (1986) |

[20] | A numerical method for solving the equations of compressible. Viscous flow. AIAA Paper 81-0110, 1981. |

[21] | Analysis of a local matrix preconditioner for the 2-D Navier-Stokes equations. AIAA Paper 93-3330, 1993. |

[22] | Three-dimensional hypersonic flow simulations with the CSCM implicit upwind Navier-Stokes method. AIAA Paper 87-1114, 1987. |

[23] | A new implicit algorithm for fluid flow. Proceedings of AIAA 13th CFD Conference, Snowmass, Colorado, 1997; 112-119. |

[24] | Jameson, International Journal of Computational Fluid Dynamics 4 pp 171– (1995) |

[25] | Jameson, International Journal of Computational Fluid Dynamics 5 pp 1– (1995) |

[26] | Warming, Mathematics of Computation 29 pp 1037– (1975) |

[27] | Abarbanel, Journal of Computational Physics 41 pp 1– (1981) |

[28] | How many steps are required to solve the Euler equations of steady, compressible flow: in search of a fast solution algorithm. AIAA Paper 2001-2673, AIAA 15th Computational Fluid Dynamics Conference, June 11-14, Anaheim, California, 2001. |

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