The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: The Crank–Nicolson scheme and the method of lines (MOL)

Abstract

MHD equations have many applications in physics and engineering. The model is coupled equations in velocity and magnetic field and has a parameter namely Hartmann. The value of Hartmann number plays an important role in the equations. When this parameter increases, using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem. In the present paper a numerical meshless method based on radial basis functions (RBFs) is provided to solve MHD equations. For approximating the spatial variable, a new approach which is introduced by Bozzini et al. (2015) is applied. The method will be used here is based on the interpolation with variably scaled kernels. The methodology of the new technique is defining the scale function c on the domain Ω⊂Rd. Then the interpolation problem from the data locations xj∈Rd transforms to the new interpolation problem in the data locations (xj,c(xj))∈Rd+1 (Bozzini et al., 2015). The radial kernels used in the current work are Multiquadrics (MQ), Inverse Quadric (IQ) and Wendland’s function. Of course the latter one is based on compactly supported functions. To discretize the time variable, two techniques are applied. One of them is the Crank–Nicolson scheme and another one is based on MOL. The numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem. Also in appendix, we provide a computational algorithm for implementing the new technique in MATLAB software.