A study of the Orr–Sommerfeld and induction equations by Galerkin and Petrov–Galerkin spectral methods utilizing Chebyshev polynomials

The article discusses two spectral methods, namely the Galerkin and the Petrov–Galerkin methods, for linear stability analysis of magneto-hydrodynamic (MHD) equations describing the flow of an electrically conducting fluid in the presence of a tangential magnetic field. The stability and spectral accuracy of both methods have been compared by examining the most unstable eigensolution of the Orr–Sommerfeld (OS) and induction equations. The Petrov–Galerkin spectral method (PGSM) used in this work has been developed by choosing function spaces and basis functions that always lead to banded coefficient matrices. The Galerkin spectral method (GSM), on the contrary, leads to dense matrices when Chebyshev polynomials are utilized in a weighted inner product space. We have found that both the GSM and the PGSM can produce results with minimal round-off errors, as confirmed by computing the most unstable eigenvalue of the OS equations (Re $=10^4$) to 14 decimal places of accuracy in double precision. We show that with properly scaled basis functions the GSM leads to coefficient matrices with bounded condition numbers, both for the OS equation and for the coupled OS and induction equations. This allows to achieve accurate results with double precision for any number of $N$ for both the GSM and the PGSM. The analysis of the different behavior of the condition numbers suggests that the proposed two methods, based on the Chebyshev polynomials, can become a useful computer-based tool that is capable of finding a numerical solution to both the hydrodynamic and the MHD equations at very high Reynolds numbers.