Boundary formulations for high-order finite differences on staggered meshes

Discretizations of the compressible flow equations with flow variables defined at different staggered positions on a regular mesh have accuracy and stability advantages over standard collocated discretization, but implementation of boundary conditions is hampered without all flow variables available at any boundary point. Boundary schemes for implementation of the boundary conditions compatible with staggered mesh discretizations are considered in this study. We focus on a combination of fourth- and fifth-order schemes near the boundary that are stable when sixth-order centered schemes are used for the interior points. Characteristics-based formulations provide a physically meaningful treatment for all the variables, avoiding the use of extrapolation. However its application on staggered meshes has not been systematically studied and its implementation is unclear since this method requires collocation of all the variables at the boundary, which is not natural for standard staggered mesh formulations. We show that including all the flow variables at the boundary can be done in a way that does not affect resolution or accuracy of the formulation. Predictions based upon analysis with model equations are verified with a staggered mesh flow solver.