Two new upwind difference schemes for a coupled system of convection–diffusion equations arising from the steady MHD duct flow problems

Abstract

In this paper, we develop two new upwind difference schemes for solving a coupled system of convection–diffusion equations arising from the steady incompressible MHD duct flow problem with a transverse magnetic field at high Hartmann numbers. Such an MHD duct flow is convection-dominated and its solution may exhibit localized phenomena such as boundary layers, namely, narrow boundary regions where the solution changes rapidly. Most conventional numerical schemes cannot efficiently solve the layer problems because they are lacking in either stability or accuracy. In contrast, the newly proposed upwind difference schemes can achieve a reasonable accuracy with a high stability, and they are capable of resolving high gradients near the layer regions without refining the grid. The accuracy of the first new upwind scheme is O( h + k) and the second one improves the accuracy to O( ε 2( h + k) + ε( h 2 + k 2) + ( h 3 + k 3)), where 0 < ε ≔ 1/ M ≪ 1 and M is the high Hartmann number. Numerical examples are provided to illustrate the performance of the newly proposed upwind difference schemes.