Wavelet Optimized Adaptive Mesh for MHD Flow Problems


There are many problems in science and engineering where the solution shows a boundary layer character. Near the boundary the gradient is large in contrast with the smooth behaviour in the central core. A uniform grid is, therefore, not suitable for a numerical solution. MHD flow problems belong to this category where a velocity and induced magnetic field profiles get flattened in a transverse flow. In the present paper an optimized grid has been generated using interpo-lating wavelets. The results are compared with those obtained using uniform grid, the finite element method and also from the analytical solution.

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B. Singh, A. Bhardwaj and R. Ali, "Wavelet Optimized Adaptive Mesh for MHD Flow Problems," Applied Mathematics, Vol. 3 No. 2, 2012, pp. 127-134. doi: 10.4236/am.2012.32020.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. Kolin, “An Electromagnetic Flowmeter—The Principle and Its Applications to Blood Flow Measurement,” Proceedings of the Society for Experimental Biology and Medicine, Vol. 35, 1936, pp. 53-56.
[2] A. Kolin, “Electromagnetic Blood Flow Meters,” Science, Vol. 130, No. 3382, 1959, pp. 1088-1097. doi:10.1126/science.130.3382.1088
[3] J. A. Jacobs, Ed., “Geomagnetism I and II,” Academic Press, Cambridge, 1987.
[4] J. A. Shercliff, “Steady Motion of Conducting Fluids in Pipes under Transverse Magnetic Fields,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 49, No. 1, 1953, pp. 136-144. doi:10.1017/S0305004100028139
[5] C. C. Changa and T. S. Lundgren, “Duct Flow in Magnetohydrodynamics,” Zeitschrift für Angewandte Mathematik und Physik (ZAMP), Vol. 12, No. 2, 1961, pp. 100-114.
[6] J. C. R. Hunt, “Magnetohydrodynamic Flow in Rectangular Duct,” Journal of Fluid Mechanics, Vol. 21, No. 4, 1965, pp. 577-590. doi:10.1017/S0022112065000344
[7] B. Singh and J. Lal, “Finite Element Method in Magnetohydrodynamic Channel Flow Problems,” International Journal for Numerical Methods in Engineering, Vol. 18, No. 7, 1982, pp. 1104-1111. doi:10.1002/nme.1620180714
[8] B. Singh and J. Lal, “Finite Element Method for Unsteady MHD Flow through Pipes with Arbitrary Wall Conducting,” International Journal for Numerical Methods in Fluids, Vol. 4, No. 3, 1984, pp. 291-302. doi:10.1002/fld.1650040307
[9] B. Singh, A. Bhardwaj and R. Ali, “A Wavelet Method for Solving Singular Integral Equation of MHD,” Applied Mathematics & Computation, Vol. 214, No. 1, 2009, pp. 271-279. doi:10.1016/j.amc.2009.03.075
[10] D. L. Donoho, “Interpolating Wavelet Transform,” Technical Report, Stanford University, Palo Alto, 1992.
[11] A. Harten, “Adaptive Multiresolution Schemes for Shock Computations,” Journal of Computational Physics, Vol. 115, No. 2, 1994, pp. 319-338. doi:10.1006/jcph.1994.1199
[12] L. Jameson, “A Wavelet Optimized Very High Order Adaptive Grid and Numerical Method,” SIAM Journal on Scientific Computing, Vol. 19, No. 6, 1998, pp. 19802013. doi:10.1137/S1064827596301534
[13] O. V. Vasilyev and C. Bowman, “Second Generation Wavelet Collocation Method for Solution of Partial Differential Equations,” Journal of Computational Physics, Vol. 165, No. 2, 2000, pp. 660-693. doi:10.1006/jcph.2000.6638
[14] V. Kumar and M. Mehra, “Wavelet Optimized Finite Difference Method Using Interpolating Wavelets for Self Adjoint Singularly Perturbed Problems,” Journal of Computational and Applied Mathematics, Vol. 230, No. 2, 2009, pp. 803-812.
[15] I. Fatkulin and J. S. Hesthaven, “Adaptive High-Order Finite Difference Method for Nonlinear Wave Problems,” Journal of Scientific Computing, Vol. 16, No. 1, 2001, pp. 44-67. doi:10.1023/A:1011198413865
[16] Z.-L. Pei, L.-Y. Fu, G.-X. Yu and L.-X. Zhang, “A Wavelet-Optimized Adaptive Grid Method for Finite-Difference Simulation of Wave Propagation,” Bulletin of the Seismological Society of America, Vol. 99, No. 1, 2009, pp. 302-313. doi:10.1785/0120080002
[17] J. Hartmann, “Hg-Dynamics I—Theory of Laminar Flow of an Electrically Conducting Liquid in a Homogeneous Magnetic Field,” Kongelige Danske Videnskabernes Selskab. Mathematisk-fysiske Meddelelser, Copenhagen, Vol. 15, No. 6, 1937.
[18] R. R. Gold, “Magnetohydrodynamic Pipe Flow—Part I,” Journal of Fluid Mechanics, Vol. 13, No. 4, pp. 505-512. doi:10.1017/S0022112062000889

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