Chebyshev-$$\tau$$ method for certain generalized eigenvalue problems occurring in hydrodynamics: a concise survey

The aim of this survey is to give a concise but technical and, as much as possible, comprehensive introduction to the resolution of certain eigenvalue problems occurring in the research field of hydrodynamics via the Chebyshev-τ method. While many details on the construction of mathematical models (for which we will refer to notable and well-known references as reported by Chandrasekhar (Hydrodynamic and hydromagnetic stability, Dover, London, 1981); Straughan (The energy method, stability, and nonlinear convection, Springer, New York, 2004); Nield and Bejan (Convection in porous media, Springer, New York, 2017)) will not be given, much attention will be paid to the practical and theoretical aspects of the discretization of the continuum problem. Chebyshev polynomials will be employed to expand solutions of the differential eigenvalue problem and end up with a discrete eigenvalue problem. Finally, MATLAB codes for the considered problems are shown in detail and available on GitHub.