Exponential high-order compact scheme on nonuniform grids for the steady MHD duct flow problems with high Hartmann numbers

In this article, we establish an exponential high-order compact (EHOC) difference scheme on non-uniform grids for the solution of the coupled equations representing the steady incompressible, viscous magnetohydrodynamic (MHD) flow through a straight channel of rectangular section. A main advantage of the non-uniform grids-based EHOC scheme is that it could use coarser mesh to capture the details within the computational domain for the MHD flow problems with high Hartmann numbers. Numerical experiments are carried out to validate the performance of the currently proposed scheme. Computation results of the MHD flow in the 2D square-channel problems are presented for Hartmann numbers ranging from 10 to 106. The numerical solutions obtained with the newly developed EHOC scheme are also compared with analytic solutions and numerical results by other available methods in the literature. All of these numerical results demonstrate that the currently proposed scheme is accurate, efficient and robust for the wide range of Hartmann numbers 10 to 106.