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Equivalence conditions between linear Lagrangian finite element and node-centred finite volume schemes for conservation laws in cylindrical coordinates

Dante de Santis 1 Gianluca Geraci 1 Alberto Guardone 2 
1 BACCHUS - Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems
Inria Bordeaux - Sud-Ouest, UB - Université de Bordeaux, CNRS - Centre National de la Recherche Scientifique : UMR5800
Abstract : A complete set of equivalence conditions, relating the mass-lumped Bubnov-Galerkin finite element (FE) scheme for Lagrangian linear elements to node-centred finite volume (FV) schemes, is derived for the first time for conservation laws in a three-dimensional cylindrical reference. Equivalence conditions are used to devise a class of FV schemes, in which all grid-dependent quantities are defined in terms of FE integrals. Moreover, all relevant differential operators in the FV framework are consistent with their FE counterparts, thus opening the way to the development of consistent hybrid FV/FE schemes for conservation laws in a three-dimensional cylindrical coordinate system. The two-dimensional schemes for the polar and the axisymmetrical cases are also explicitly derived. Numerical experiments for compressible unsteady flows, including expanding and converging shock problems and the interaction of a converging shock waves with obstacles, are carried out using the new approach. The results agree fairly well with one-dimensional simulations and simplified models.
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Dante de Santis, Gianluca Geraci, Alberto Guardone. Equivalence conditions between linear Lagrangian finite element and node-centred finite volume schemes for conservation laws in cylindrical coordinates. International Journal for Numerical Methods in Fluids, Wiley, 2014, 74 (7), pp.514-542. ⟨10.1002/fld.3862⟩. ⟨hal-00935561⟩

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