A Tailored Finite Point Method for Solving Steady MHD Duct Flow Problems with Boundary Layers (Communications in Computational Physics, 2011)
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 numberis large, the MHD duct flow is convection-dominated and thus its solution may exhibitlocalized phenomena such as the boundary layer. Most conventional numericalmethods can not efficiently solve the layer problem because they are lacking in eitherstability or accuracy. However, the proposed tailored finite point method is capableof resolving high gradients near the layer regions without refining the mesh. Firstly,we devise the tailored finite point method for the scalar inhomogeneous convection-diffusionproblem, and then extend it to the MHD duct flow which consists of a coupledsystem of convection-diffusion equations. For each interior grid point of a givenrectangular mesh, we construct a finite-point difference operator at that point withsome nearby grid points, where the coefficients of the difference operator are tailoredto some particular properties of the problem. Numerical examples are provided toshow the high performance of the proposed method.