The Geometric Conservation Law - A link between finite-difference and finite-volume methods of flow computation on moving grids

Boundary-conforming coordinate transformations are used widely to map a flow region onto a computational space in which a finite-difference solution to the differential flow conservation laws is carried out. This method entails difficulties with boundary conditions, with maintenance of global conservation, and with computation of the local volume element under time-dependent mappings that result from boundary motion. To improve the method, a differential 'Geometric Conservation Law' (GCL) is formulated that governs the spatial volume element under an arbitrary mapping. The GCL is solved numerically along with the flow conservation laws using conservative difference operators. Numerical results are presented for both implicit and explicit difference schemes applied to the unsteady Navier-Stokes equations and to the steady supersonic flow equations.