Unification of Some Advection Schemes in Two Dimensions

In this paper a relationship between two approaches towards construction of genuinely two-dimensional upwind advection schemes is established. One of these approaches is of tile control volume type applicable on structured cartesian meshes. It resulted (see [14], [15]) in the compact high resolution schemes capable of maintaining second order accuracy in both homogeneous and inhomogeneous cases. Another one is the fluctuation splitting approach (see [11], [3], [12], [17]), which is well suited for triangular (and possibly) unstructured meshes. Understanding the relationship between these two approaches allows us to formulate here a new fluctuation splitting high resolution (i.e. possible use of artificial compression, while maintaining positivity property) scheme. This scheme is shown to be linearity preserving in inhomogeneous as well as homogeneous cases. *The first, author was supported by DOE grant. DE-FG02-92ER25139, while he was in residence at Courant Institute of Mathematical Sciences, 251 Mercer St., New York University, New York. NY 10012. Both authors were also supported in part by the National Aeronautics and Space Administration under NASA Contract No. N'ASI-19480 while in residence at the hlstitute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001.