Symmetric Positive Definite Subcell CVD Schemes for Cell-Centred Quadrilateral Grids

A new family of locally conservative subcell flux-continuous schemes is presented for cell-centred approximation of the general-tensor pressure equation on quadrilateral grids. The local position of flux continuity quadrature point defines the scheme. This work continues the development of symmetric positive definite subcell Control-Volume Distributed (CVD) schemes first introduced in [1, 2], where a piecewise constant general geometry-permeability tensor approximation is introduced over each subcell of a control-volume. Physical-space flux-continuous schemes possess non-symmetric matrices for general quadrilateral cells, or indeed any general cell type. The subcell tensor approximation ensures that a flux-continuous finite volume scheme is obtained with a symmetric positive definite (SPD) discretization matrix on any grid, structured or unstructured [1, 2, 3]. <br><br>By definition, tensor approximation at the subcell level leads to a finer scale tensor approximation compared to the cell level, and consequently can be expected to yield a superior SPD approximation to that of earlier cell-centred cell-wise constant SPD tensor schemes. A numerical convergence study confirms that the SPD subcell tensor approximation reduces solution errors when compared to the SPD cell-wise constant tensor schemes. In addition, constructing the subcell tensor approximation using control-volume face geometry yields the best results consistent with design of the schemes. Comparisons are also made with the physical space schemes. A particular quadrature point is found to give the best numerical convergence of the subcell schemes for the cases tested.<br><br>