## 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.

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 | 22 |

Journal | Journal of Computational Physics |

Volume | 229 |

Issue number | 3 |

Publication status | Published - 9 Sep 2010 |

### Bibliographical note

G. Tryggvason## Keywords

- Navier-Stokes
- High-order Methods
- Implicit LU-SGS algorithm
- Von Neumann analysis
- p-multigrid