Fast method and convergence analysis for the magnetohydrodynamic flow and heat transfer of fractional Maxwell fluid

Abstract

This work investigates the unsteady magnetohydrodynamic (MHD) flow and heat transfer of fractional Maxwell fluids in a square cavity, which is under the influence of the Hall effect and radiation heat. The coupled model is formed from the momentum equation based on the fractional constitutive relationship and the fractional heat-conduction equation derived from the Fourier law. The fractional coupled model is solved numerically by combining the weighted and shifted Grünwald difference method in the temporal direction with the spectral method based on Lagrange-basis polynomials in the spatial direction. In addition, we propose a fast method to reduce the computational time and the memory requirements of the actual calculation. We also prove the stability and convergence of the numerical scheme with the fast method. Furthermore, a numerical example is given to verify the efficiency of the numerical method and of the theoretical analysis. An example of non-smooth solutions is dealt with by adding correction terms. Finally, an example is considered to discuss the effects of the Hartmann number, the Hall parameter, and the thermal radiation parameter on the MHD flow and heat transfer of a fractional Maxwell fluid in a square cavity.