Spectral Methods

TL;DR: Spectral methods are used to numerically solve ODEs with boundary conditions. Functions are approximated by polynomials in a Chebychev basis. Differential operators are approximated as rectangular matrices and boundary conditions add additional rows to make them square. These matrices can be diagonalized using standard linear algebra methods.

Abstract: Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.