Novel connection of spectral scheme and one-step of s-order approaches for MHD flows enclosed a duct

A challenging and common problem that frequently arises in the fields of physics and engineering, two-dimensional (2D) incompressible, viscous MHD duct flows have significant theoretical and practical significance due to their numerous and widespread applications in astrophysics, geology, power generation, MHD generators, electromagnetic pumps, accelerators, blood flow measurements, drug delivery, and other areas. Therefore, a robust solution to such a problem becomes a challenging task for the research community. This framework develops a novel connection to inspect the accurate and rapid convergent solutions of a coupled system of convection-diffusion equations arising in 2D unsteady MHD flows. This coupling is based on one-step s-stage/order methods to approximate the temporal variable with the Vieta-Fibonacci polynomials-based spectral method to estimate the spatial variables. The spatial derivative terms given in the problem under discussion are replaced by new operational matrices of integer order. The paper incorporates related theorems to provide a mathematical validation of the techniques. Additionally, we conduct a study on convergence and error bonds to verify the computational algorithm's mathematical formulation. A thorough comparison analysis illustrates the validity, correctness, and dependability of the computational approach that is now recommended. Novel investigation includes the spectral technique coupled with the fourth-order Runge-Kutta method handles the nonlinear issue very well to investigate the exact smooth solutions to physical problems. The suggested schemes are discovered to have an exponential order of convergence in the spatial direction, and the COC in the temporal direction confirms the findings of previous research.