Abstract: In this paper, we review the recent development in our research group on the construction of high order finite volume schemes on unstructured grids with compact stencils. The second order finite volume schemes have been applied extensively in commercial and in-house Computational Fluid Dynamics (CFD) software. When one tries to further increase the order of accuracy of the finite volume schemes, the large reconstruction stencil will be an inevitably encountered problem and is considered as the main technical bottleneck of the high order finite volume schemes. In recent years, the high order reconstruction algorithms on the compact stencils have been systematically studied in our group. A new concept called "operational compact" is proposed that provides new possibilities to design compact reconstruction algorithms. Using this concept, three new high order accurate reconstruction algorithms, namely compact least squares reconstruction, variational reconstruction, and multi-step reconstruction, are proposed and implemented. The common feature of the proposed algorithms is that arbitrarily high order polynomial reconstructions can be realized on the compact stencils. These reconstruction algorithms are reviewed in detail in the present paper. Further insights into the basic ideas, implementation details and future developments are also provided. The variational reconstruction will be further improved and applied under the support of the National Numerical Windtunnel (NNW) Project.

Key words: high order accurate scheme, finite volume scheme, unstructured grid, compact stencil, reconstruction algorithm, National Numerical Windtunnel (NNW) Project