Numerical approximation for MHD flows of generalized viscoelastic fluid

The numerical solution of MHD flow of fractional viscoelastic fluid is considered in this article. The main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the fractional order differential equation. A schema combining a finite difference approach in time direction and spectral approximations in the space direction is proposed and analyzed. A detailed analysis shows that the proposed scheme is unconditionally stable. Stability and convergence of the method are rigorously established, and we prove that the convergent order is , where , N and m are respectively time step size, polynomial degree, and regularity in the space variable, and is the fractional derivative of the exact solution. Numerical computations are shown which demonstrate the effectiveness of the method and confirm the theoretical results. At last, the influence of fractional order and the magnetic effect on the solution is discussed.