Exponential compact ADI method for a coupled system of convection-diffusion equations arising from the 2D unsteady magnetohydrodynamic (MHD) flows

In this paper, an exponential high-order compact alternating direction implicit (EHOC-ADI) difference method is developed for solving the coupled equations representing the unsteady incompressible, viscous magnetohydrodynamic (MHD) flow through a straight pipe of rectangular section. The method, in which the Crank-Nicolson scheme is used for the time discretization and an original EHOC difference scheme established for the steady 1D coupled system of convection-diffusion equations is used for the spatial discretization, is second order accurate in time and fourth-order accurate in space and requires only a regular five-point 2D stencil similar to that in the standard second-order methods and the three-point stencil for each 1D operator. A distinguishing desirable property of the proposed method is combining the computational efficiency of the lower order methods with superior accuracy inherent in high order approximations. The unconditional stable character of the method is verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are carried out to illustrate and assess the performance and the accuracy of the method proposed currently. Computational results of the MHD flow in the 2D square-channel problems are presented for Hartmann numbers ranging from 0 to 106 and compared with the exact solutions and those obtained using other available methods in the literatures. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.