On the geometric conservation law for high-order discontinuous Galerkin discretizations on dynamically deforming meshes

An approach for constructing high-order Discontinuous Galerkin schemes which preserve discrete conservation in the presence of arbitrary mesh motion, and thus obey the GCL, is derived. The approach is formulated for the most general case where only the coordinates defining the mesh elements are known at discrete locations in time, and results in the prescription of higher-order quadrature rules for certain terms in the governing equations in arbitrary Lagrangian Eulerian (ALE) form. For BDF1 temporal discretizations, the approach is exactly equivalent to a space-time formulation, while providing a natural extension to more complex discretizations such as BDF2. The method is shown to preserve the temporal accuracy of the underlying time-stepping scheme, for BDF1 and BDF2 schemes, for high-order spatial discretizations of the flow equations and mesh motion definition ranging up to fifth order accurate. Future work will investigate extensions of this approach to higher-order temporal schemes such as implicit Runge-Kutta schemes.