A Compact High-Order Finite Volume Method for Computing Shock Waves on Arbitrary Grids

A cell-centered finite volume method based on a variational reconstruction, termed FV(VR) in this paper, is developed for compressible flows on 3D arbitrary grids. In this method, a linear polynomial solution is reconstructed using a newly developed variational formulation. Like the least-squares reconstruction, the variational reconstruction has the property of 1-exactness. Unlike the cell-centered finite volume method based on the least-squares reconstruction, termed FV(LS) in this paper, the resulting FV(VR) method is stable even on tetrahedral grids, since its stencils are intrinsically the entire mesh. However, the data structure required by FV(VR) is the same as FV(LS) and is thus compact and simple. A nonlinear WENO reconstruction is used to suppress nonphysical oscillations in the vicinity of strong discontinuities. A variety of the benchmark test cases are presented to assess the accuracy, efficiency, robustness, and flexibility of this finite volume method. The numerical experiments indicate that the developed FV(VR) method is able to maintain the linear stability, attain the designed second order of accuracy, and outperform the FV(LS) method without a significant increase in computing costs and storage requirements.