Commun. Comput. Phys., 10 (2011), pp. 161-182. |
A Tailored Finite Point Method for Solving Steady MHD Duct Flow Problems with Boundary Layers Po-Wen Hsieh ^{1}, Yintzer Shih ^{2}, Suh-Yuh Yang ^{1*} 1 Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan.2 Department of Applied Mathematics, National Chung Hsing University, Taichung 40227, Taiwan. Received 7 January 2010; Accepted (in revised version) 2 July 2010 Available online 30 March 2011 doi:10.4208/cicp.070110.020710a Abstract In this paper we propose a development of the finite difference method, called the tailored finite point method, for solving steady magnetohydrodynamic (MHD) duct flow problems with a high Hartmann number. When the Hartmann number is large, the MHD duct flow is convection-dominated and thus its solution may exhibit localized phenomena such as the boundary layer. Most conventional numerical methods can not efficiently solve the layer problem because they are lacking in either stability or accuracy. However, the proposed tailored finite point method is capable of resolving high gradients near the layer regions without refining the mesh. Firstly, we devise the tailored finite point method for the scalar inhomogeneous convection-diffusion problem, and then extend it to the MHD duct flow which consists of a coupled system of convection-diffusion equations. For each interior grid point of a given rectangular mesh, we construct a finite-point difference operator at that point with some nearby grid points, where the coefficients of the difference operator are tailored to some particular properties of the problem. Numerical examples are provided to show the high performance of the proposed method. AMS subject classifications: 65N06, 65N12, 76W05Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Magnetohydrodynamic equations, Hartmann numbers, convection-dominated problems, boundary layers, tailored finite point methods, finite difference methods. *Corresponding author. Email: 952401001@cc.ncu.edu.tw (P.-W. Hsieh), yintzer_shih@email.nchu.edu.tw (Y. Shih), syyang@math.ncu.edu.tw (S.-Y. Yang) |