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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.

A rational high-order compact (RHOC) finite difference (FD) method on the nine-point stencil is proposed for solving the steady-state two-dimensional NavierStokes equations in the stream functionvorticity form. The resulting system of algebra equations can be solved by using the point-successive over- or under-relaxation (SOR) iteration. Numerical experiments, involving two linear and two nonlinear problems with their analytical solutions and two flow problems including the lid driven cavity and backward-facing step flows, are carried out to validate the performance of the newly proposed method. Numerical solutions of the driven cavity problem with different grid mesh sizes (maximum being 513513) for Reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature.

In the majority of research on incompressible magnetohydrodynamic (MHD) flows, the simplified model with the low magnetic Reynolds number assumption has been adopted because it reduces the number of equations to be solved. However, because the effect of flow on magnetic field is also neglected, the solutions of the simplified model may be different from those of the full model. As an example, the flow of an electrically conducting fluid past a circular cylinder under a magnetic field is investigated numerically using the simplified and full models in this paper. To solve the problems, two second-order compact finite difference algorithms based on the streamfunction-velocity formulation of the simplified model and the quasi-streamfunction-velocity formulation of the full model are developed respectively. Numerical simulations are carried out over a wide range of Hartmann number for steady-state laminar problems with both models. For the full model, magnetic Reynolds number (Rem) is chosen from 0.01 to 10. The computed results show that solutions of the simplified MHD model are not exactly the same as those of the full MHD model for this flow problem in most cases even if Rem in the full model is very low. Only in the special case that a strong external magnetic field is exerted perpendicular to the dominant flow direction, can the simplified MHD model be regarded as an approximation of the full MHD model with low Rem.