On the Temporal and Spatial Accuracy of Spectral Difference Method on Moving Deformable Grids and the Effect of Geometric Conservation Law

In this paper we studied and examined the formulation of conservation laws on dynamic moving deforming meshes, in the context of the Geometric Conservation Law (GCL) compliant high order temporal and spatial methods. In particular, we implemented the high order explicit Runge-Kutta (ERK) as the time integrator, and the Spectral Difference (SD) method for the spatial discretization. In order to apply SD method to moving boundary problems, a strategy for moving and deforming the mesh need to be devised. A strategy considered here employs time-dependent analytical coordinate transformation. While this method is simple, without the need for remeshing the flow domain, questions remain whether such a strategy would compromise the spatial and temporal accuracy of the high order differencing scheme and the high order time integration scheme. To investigate this, numerical experiments have been carried out to demonstrate the temporal and spatial accuracy of the new SD-ERK scheme in dynamic deformable flow domains. This study shows that the designated SD schemes with explicit Runge-Kutta method preserves the high order accurate nature of the original methods in moving deforming meshes. With regard to the effect of the Geometric Conservation Law on the temporal accuracy, spatial accuracy, freestream preservation, discretized conservation, and non-linear stabilty of the high order methods, our study shows that a properly designed GCL-compliant formulation will not affect the underlying temporal and spatial accuracy of the original methods, while at the same time preserve the freestream constant solution. Hence it makes the solver more conservative and accurate. The elmination of the source term due to mesh volume element change could potentially improve the non-linear stability of the solver, but further work is needed for a more concrete assessment on the effect of non-lnear stability.