Vertex-centroid finite volume scheme on tetrahedral grids for conservation laws

Vertex-centroid schemes are cell-centered finite volume schemes for conservation laws which make use of both centroid and vertex values to construct high-resolution schemes. The vertex values must be obtained through a consistent averaging (interpolation) procedure while the centroid values are updated by the finite volume scheme. A modified interpolation scheme is proposed which is better than existing schemes in giving positive weights in the interpolation formula. A simplified reconstruction scheme is also proposed which is also more efficient and leads to more robust schemes for discontinuous problems. For scalar conservation laws, we develop limited versions of the schemes which are stable in maximum norm by constructing suitable limiters. The schemes are applied to compressible flows governed by the Euler equations of inviscid gas dynamics.