Further studies on Geometric Conservation Law and applications to high-order finite difference schemes with stationary grids

The metrics and Jacobian in the fluid motion governing equations under curvilinear coordinate system have a variety of equivalent differential forms, which may have different discretization errors with the same difference scheme. The discretization errors of metrics and Jacobian may cause serious computational instability and inaccuracy in numerical results, especially for high-order finite difference schemes. It has been demonstrated by many researchers that the Geometric Conservation Law (GCL) is very important for high-order Finite Difference Methods (FDMs), and a proper form of metrics and Jacobian, which can satisfy the GCL, can considerably reduce discretization errors and computational instability. In order to satisfy the GCL for FDM, we have previously developed a Conservative Metric Method (CMM) to calculate the metrics [1] and the difference scheme δ3δ3 in the CMM is determined with the suggestion δ3=δ2δ3=δ2. In this paper, a Symmetrical Conservative Metric Method (SCMM) is newly proposed based on the discussions of the metrics and Jacobian in FDM from geometry viewpoint by following the concept of vectorized surface and cell volume in Finite Volume Methods (FVMs). Interestingly, the expressions of metrics and Jacobian obtained by using the SCMM with second-order central finite difference scheme are equivalent to the vectorized surfaces and cell volumes, respectively. The main advantage of SCMM is that it makes the calculations based on high-order WCNS schemes aroud complex geometry flows possible and somewhat easy. Numerical tests on linear and nonlinear problems indicate that the quality of numerical results may be largely enhanced by utilizing the SCMM, and the advantage of the SCMM over other forms of metrics and Jacobian may be more evident on highly nonuniform grids.