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 Geometric Conservation Law (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 arbitrary geometrically high-order curved mesh element deformation is considered. The method is applied to the governing equations in arbitrary Lagrangian Eulerian (ALE) form, and results in a prescription for computing integrated grid speed terms along with the requirement of higher-order quadrature rules in both space and time. For a first-order backward difference time-integration scheme (BDF1), the approach is exactly equivalent to a space–time formulation, while providing a natural extension to more complex discretizations such as high-order backwards difference schemes, Crank–Nicholson schemes, and implicit Runge–Kutta (IRK) methods. Numerical results are performed using up to fifth-order accuracy in space and fourth-order accuracy in time, and the design accuracy of the underlying time-stepping scheme is shown to be preserved in the presence of arbitrary curved-element mesh motion.