On the admissibility of stencils for first order polynomial reconstruction on three-dimensional unstructured meshes

Abstract

Starting from the work of Abgrall [1], several authors proposed finite volume methods for the solution of fluid dynamics problems on unstructured meshes, with order of spatial accuracy greater than two and possessing non-oscillatory properties. A recent review of these methods can be found in [2]. Also Sonar [3] gives formal proofs of several properties of non-oscillatory schemes on unstructured meshes. A technique that allows obtaining spatial order of accuracy three or four was developed by Hu and Shu [4]. On this weighted essentially non-oscillatory scheme, first order-polynomials are used to obtain reconstructions of higher order spatial accuracy. Thus, admissibility of the corresponding stencils is an important question to be addressed in considering possible extensions to the 3D case. In two dimensions, the equivalence between admissibility of a three control volume stencil and collinearity of the mass centers is a known fact [5]. In this work we prove that admissibility of a three-dimensional four control volumes stencil is equivalent to the fact of the mass centers not being coplanar. The technique of proof can easily be adapted to the two-dimensional case for the analogous result as well as for proving admissibility in higher order spaces without difficulties.