Superconvergence of Characteristics Marker and Cell Scheme for the Navier--Stokes Equations on Nonuniform Grids

In this paper, a characteristics marker and cell (C-MAC) scheme is established for the Navier--Stokes equations on nonuniform grids. Error estimates for the pressure and velocity in different discrete norms are established rigorously and carefully. We obtain the second order superconvergence in the discrete $L^2$ norm for both velocity and pressure and the second order superconvergence for some terms of the $H^1$ norm of the velocity for the C-MAC scheme on nonuniform grids, which have not been reported before. Finally, some numerical experiments are presented to show the correctness and accuracy of the C-MAC scheme and the robustness and efficiency of the overall solution technique has been demonstrated using the lid-driven cavity model.