The design and application of upwind schemes on unstructured meshes

Order of accuracy study of unstructured grid finite volume upwind schemes

“…This difference is associated with the gradient calculation, which is computed in the standard finite volume fashion in which the derivatives are transformed into integrals along the boundaries of the control volume. Hence, in one approach, the control volumes used for the cell averaged gradient calculations are the triangles themselves, while the other approach uses an extended control volume (Barth and Jespersen, 1989).…”

“…The first approach consists in defining V Ω = V i , i.e., the triangles themselves are used as the control volumes for the gradient calculation. This is the simplest approach possible, but it is usually criticized in the literature (Barth and Jespersen, 1989) because it cannot recover the correct gradient of a linear function.…”

“…In particular, the emphasis is on triangular grids and flux-vector splitting schemes. Hence, a scalar version of Liou's AUSM + scheme (Liou, 1996) is presented both as a nominally 1st-order scheme and as a nominally 2nd-order scheme. The 2nd-order version uses a MUSCL reconstruction (van Leer, 1979), and the reconstruction process options implemented are discussed in detail in the forthcoming sections.…”

“…1 Since the pioneering work of Barth and Jespersen (1989), various upwind, nominally second order accurate, finite volume schemes have been proposed in the literature (see, for instance, Durlofsky, Engquist and Osher, 1992;Lin, Wu and Chin, 1993;Venkatakrishnan, 1995;Aftosmis, Gaitonde and Tavares, 1995;Sleigh et al, 1998; Figueira da Silva, Azevedo and Korzenowski, 1999). In these papers, spatial second order accuracy is sought by some form of gradient evaluation within the control volume, followed by extrapolation of the cell centered properties up to the cell interfaces.…”

“…The present work presents a detailed numerical study of the order of accuracy of some proposed cell centered, finite volume schemes typically used for the solution of the 2-D gasdynamic equations on triangular unstructured grids (Barth and Jespersen, 1989; Figueira da Silva, Azevedo and Korzenowski, 1999). The schemes studied are nominally second order accurate, based on a Paper accepted August, 2009.…”