Comptes Rendus
Verification of higher-order discontinuous Galerkin method for hexahedral elements
Comptes Rendus. Mécanique, Volume 333 (2005) no. 9, pp. 719-725.

A high-order implementation of the Discontinuous Galerkin (dg) method is presented for solving the three-dimensional Linearized Euler Equations on an unstructured hexahedral grid. The method is based on a quadrature free implementation and the high-order accuracy is obtained by employing higher-degree polynomials as basis functions. The present implementation is up to fourth-order accurate in space. For the time discretization a four-stage Runge–Kutta scheme is used which is fourth-order accurate. Non-reflecting boundary conditions are implemented at the boundaries of the computational domain.The method is verified for the case of the convection of a 1D compact acoustic disturbance. The numerical results show that the rate of convergence of the method is of order p+1 in the mesh size, with p the order of the basis functions. This observation is in agreement with analysis presented in the literature.

L'implantation d'une méthode de Galerkine discontinue d'ordre élevé est présentée pour résoudre les équations d'Euler linéarisées tridimensionnelles en maillage non structuré avec des éléments hexaédriques. La méthode est basée sur l'utilisation de formules de quadrature non définies à l'avance et l'ordre élevé de la méthode est obtenu en utilisant des polynômes de degré élevé comme fonctions de base. La technique implantée est précise jusqu'à l'ordre 4 en espace. Pour la discrétisation en temps une méthode de Runge–Kutta précise à l'ordre 4 est utilisé. Des conditions aux limites non réfléchissantes sont implantées aux frontières du domaine de calcul. La méthode est validée sur le cas 1D d'une perturbation acoustique. Les résultats numériques montrent que le taux de convergence de la méthode est d'ordre p, p étant l'ordre des fonctions de base. Ce résultat est en accord avec les analyses présentées dans la littérature.

Published online:
DOI: 10.1016/j.crme.2005.07.012
Keywords: Acoustics, Computational aeroacoustics, Discontinuous Galerkin method, Finite element method, Hexahedral elements
Mot clés : Acoustique, Aéroacoustique numérique, Méthode de Galerkine discontinue, Méthode des éléments finis, Éléments hexaédriques

Hüseyin Özdemir 1; Rob Hagmeijer 1; Hendrik Willem Marie Hoeijmakers 1

1 Department of Mechanical Engineering, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands
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Hüseyin Özdemir; Rob Hagmeijer; Hendrik Willem Marie Hoeijmakers. Verification of higher-order discontinuous Galerkin method for hexahedral elements. Comptes Rendus. Mécanique, Volume 333 (2005) no. 9, pp. 719-725. doi : 10.1016/j.crme.2005.07.012. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2005.07.012/

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