Multigrid formulation of polynomial flux-difference splitting for steady Euler equations.

*(English)*Zbl 0711.76069Summary: A flux-difference splitting method based on the polynomial character of the flux vectors is applied to steady Euler equations. The discretization is done with a vertex-centered finite volume method. In first-order form, a discrete set of equations is obtained which is both conservative and positive. The flux-difference splitting is done in an algebraically exact way, so that shocks are represented without wiggles. Due to the positivity, the set of equations can be solved by collective relaxation methods. A full multigrid method based on symmetric successive relaxation, full weighting, bilinear interpolation and W-cycle is presented. In first-order form, typical full multigrid efficiency is achieved. This is demonstrated on the GAMM transonic bump test-case. The second-order formulation is based on the Roe-Chakravarthy minmod-limiter. The discrete system is solved using a multigrid defect-correction formulation. The second-order formulation is demonstrated on Harten’s shock reflection problem.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76L05 | Shock waves and blast waves in fluid mechanics |

76H05 | Transonic flows |

##### Keywords:

flux-difference splitting method; steady Euler equations; symmetric successive relaxation, full weighting, bilinear interpolation; W-cycle; GAMM transonic bump test-case; Roe-Chakravarthy minmod-limiter; Harten’s shock reflection problem
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