Characterizing magnetohydrodynamical and induced magnetic effects inside a square channel using an inventive combination of spectral and one-step techniques

A difficult matter for researchers is to develop numerical methods to analyze the behavior of nonlinear dynamical systems that arise in mathematical physics due to their nonlinearity and multi-dimensional nature. In this affection, a novel and effective coupling of spectral and one-step approaches is suggested and used to inspect the robust solutions to the mathematical problems. This proposed method is grounded on the one-step sixth-order Runge–Kutta (RK) method and the shifted Vieta–Fibonacci (sVF) polynomial-based spectral approach. First, use the sixth-order one-step technique for the temporal approximation, then utilize the sVF polynomials to estimate the spatial variables. The sVF polynomial-based new operational matrices are used to calculate the derivative terms of the mathematical models. Related theorems are presented in the paper to provide a mathematical validation of the technique. The suggested semi-spectral techniques transform the investigated nonlinear models into a set of linear algebraic equations, making them simpler to solve. We conduct theoretical and numerical research into the stability, convergence, and error-bound analysis to support the computational algorithm's mathematical formulation. Several mathematical models, including the Hartmann numbers ranging from 0 to 106, are taken into consideration to demonstrate the precision and efficacy of the proposed computational technique. A thorough comparison analysis illustrates the recommended computational approach's validity, correctness, and dependability. The suggested construction of sixth-order in the temporal direction and exponential order around 3.5545e0.1515M in the spatial direction is determined to be exceptional in handling nonlinear problems and examining the exact solutions.