## Abstract

The fluid dynamic equations are discretized by a high-order spectral volume (SV) method on unstructured tetrahedral grids. We solve the steady state equations by advancing in time using a backward Euler (BE) scheme. To avoid the inversion of a large matrix we approximate BE by an implicit lower-upper symmetric Gauss-Seidel (LU-SGS) algorithm. The implicit method addresses the stiffness in the discrete Navier-Stokes equations associated with stretched meshes. The LU-SGS algorithm is then used as a smoother for a p-multigrid approach. A Von Neumann stability analysis is applied to the two-dimensional linear advection equation to determine its damping properties. The implicit LU-SGS scheme is used to solve the two-dimensional (2D) compressible laminar Navier-Stokes equations. We compute the solution of a laminar external flow over a cylinder and around an airfoil at low Mach number. We compare the convergence rates with explicit Runge-Kutta (E-RK) schemes employed as a smoother. The effects of the cell aspect ratio and the low Mach number on the convergence are investigated. With the p-multigrid method and the implicit smoother the computational time can be reduced by a factor of up to 5-10 compared with a well tuned E-RK scheme.

Original language | English |
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Pages (from-to) | 828-850 |

Number of pages | 23 |

Journal | Journal of Computational Physics |

Volume | 229 |

Issue number | 3 |

DOIs | |

State | Published - 1 Feb 2010 |

## Keywords

- High-order methods
- Implicit LU-SGS algorithm
- Navier-Stokes
- Von Neumann analysis
- p-Multigrid