A new analysis of Galerkin Legendre spectral methods for coupled hyperbolic/parabolic system arising in unsteady MHD flow of Maxwell fluid and numerical simulation

Abstract

In this paper, a high-order accurate numerical method for two-dimensional coupled hyperbolic and parabolic partial differential system arising in unsteady magnetohydrodynamics (MHD) flow of upper-convected Maxwell fluid is presented. We apply Galerkin-Legendre spectral method for discretizing spatial derivatives for MHD-M (MHD flow of Maxwell fluid) equations and then Crank-Nicolson scheme is used for time derivatives. Optimal a priori error bound is derived in the L2 and H1 norm for the semidiscrete formulation. We then proceed to discuss an instability mechanism in polynomial spectral methods and show that filtering suffices to ensure stability. Optimal error bound is derived between the semi-analytical solutions and filtered semi-analytical solutions. The results are illustrated by computational experiments. Furthermore, these results indicate that the proposed method is simple, accurate, efficient, and stable for solving various MHD-M and MHD flow problems or in general coupled hyperbolic-parabolic partial differential equations.