Wavelet optimized finite difference mesh for MHD flow in a circular duct

There is a large class of important problems in scientific and engineering applications where the solution shows very little variation in the core region but very rapid changes close to the boundary of the domain of interest. Magnetohydrodynamic flow problems belong to this category. Treatment of such problems numerically, therefore, requires a non-uniform mesh, which is very fine near the boundary. The determination of the size of the grid in different parts of the domain is, therefore, a challenging problem. Some recent studies suggest that wavelet methods can be effectively applied to create an optimized mesh. The method has mostly been used for one dimensional problems. The present paper investigates the problem for fully developed MHD duct flow problems which are basically two dimensional. The boundary layer character is more pronounced in one direction as compared to the other. The governing equations comprise a coupled system of two partial differential equations with Dirichlet boundary conditions. Wavelet based adaptive mesh is first created and then the system is discretized over this mesh. Computations have been carried out at different precision levels for various Hartmann numbers for a pipe of circular section. Contours are also drawn to depict the core and the boundary layer region.