A survey of geometry conservation law for high-order finite difference method

Focusing on the geometry conservation law (GCL) for high-order finite difference schemes, the domestic and overseas studies regarding discretized geometric conservation law are reviewed systematically. Beginning with the freestream preservation phenomenon simulated by finite difference schemes, our recent studies about the geometric conservation law are summarized, including the conservative metric method (CMM) and the symmetrical conservative metric method (SCMM). Moreover, it is confirmed by several typical tests that the numerical simulations of high-order finite difference methods can be improved by satisfying the geometric conservation law. The systematic review about the geometric conservation law leads to the following conclusions: 1) The geometric conservation law cannot be satisfied by finite difference methods with the discretized metrics based on the traditional computational form. The conservative computational form of metrics together with the CMM condition should be adopted to satisfy the geometric conservation law; 2) The SCMM is a sufficient condition to satisfy the geometric conservation law, with an additional condition that the metrics and Jacobian must be discretized on the basis of the symmetrical conservative form, which are both unique; 3) The freestream preservation behavior is merely a representation of the satisfaction of the geometric conservation law; 4) The satisfaction of the geometric conservation law can effectively enhance the ability of numerical simulations with high-order finite difference schemes.