Superconvergence of MAC Scheme for a Coupled Free Flow-Porous Media System with Heat Transport on Non-uniform Grids

In this paper, a marker and cell (MAC) scheme is constructed for the free flow-porous media system with heat transport on non-uniform grids, where the Stokes-Darcy equation is employed to describe the free-flow and the porous regions. Error estimates for the velocity, pressure and temperature in different discrete norms are established rigorously and carefully by constructing discrete auxiliary functions. We obtain the second order superconvergence in the discrete L-2 norm for velocity, pressure and temperature and the second order superconvergence for some terms of the H-1 norm for the velocity on non-uniform grids. Several numerical examples verifying the theoretical results and illustrating the capabilities of the method are demonstrated.

Automated summary

This paper presents a marker and cell (MAC) scheme for the free flow-porous media system with heat transport on non-uniform grids, utilizing the Stokes-Darcy equation to describe the flow in both free-flow and porous regions. Rigorous error estimates for velocity, pressure, and temperature in different discrete norms are established through the construction of discrete auxiliary functions. The study demonstrates second-order superconvergence in the discrete L-2 norm for velocity, pressure, and temperature, as well as second-order superconvergence for certain terms of the H-1 norm for velocity on non-uniform grids.