On the discretization of spatial metrics satisfying the GCL identities

In this research, the spatial metrics used in the finite difference scheme for curvilinear coordinate system are discussed from the viewpoint of geometric interpretation. We summarized all the evaluation technique for second-order metrics proposed in the previous studies, and explain their geometric interpretation if they have appropriate geometries. Adoptingthe regular second-order finite difference scheme, only the symmetric conservative formulation has the appropriate geometry in the viewpoint of finite volume scheme. This form straightforwardly extended to the high order scheme, having appropriate geometry interpretations. This form improves the robustness of the high-order computation on the highly skewed grids compared with the asymmetric conservative formulation which is often used in the fluidic computation.