A new auxiliary volume-based gradient algorithm for triangular and tetrahedral meshes

Abstract

To have a second-order accurate centroid-based finite volume method, the gradients at the cell-centroids need to be evaluated with at least first-order accuracy. This again requires the cell face-center solution values to be interpolated with second-order accuracy when using Green-Gauss methods. It is not always possible to obtain the second-order accuracy in face interpolations with simple averaging or linear interpolation by Traditional Green-Gauss methods when the meshes are irregular in nature. In this paper, a simple and novel way to calculate first-order accurate gradients on any tri and tetra type meshes is proposed. The new method, referred to as the Circle Green-Gauss (CGG) reconstruction, is based on constructing an auxiliary volume around the cell of interest from the neighboring cells. The solution values at the faces of the constructed auxiliary volume of the cell can then be easily estimated to second-order accuracy, thereby making the gradients first-order accurate. Qualitative comparison of the proposed method with Traditional Green-Gauss (TGG) reconstruction and the Least-Squares (LS) reconstruction (used in most commercial solvers) is shown on different mesh topologies that are irregular in nature. The stability and applicability of CGG are discussed in detail through representative test problems including the method of manufactured solutions that demonstrate that the gradients are first-order accurate (so that the overall method is second-order accurate) on triangular and tetrahedral meshes.