Stable Spectral Methods for Time-Dependent Problems and the Preservation of Structure

Abstract

This paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More specifically, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonormal polynomials). We consider a simple construction of such systems and pursue its ramifications. In general, given any \(\text {C}^1(a,b)\) weight function such that \(w(a)=w(b)=0\), we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case \(a=-\infty \), \(b=+\infty \), only few powers of that matrix are bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function \(x^\alpha \textrm{e}^{-x}\) for \(x>0\) and \(\alpha >0\) and the ultraspherical weight function \((1-x^2)^\alpha \), \(x\in (-1,1)\), \(\alpha >0\), and establish their properties. Both weights share a most welcome feature of separability, which allows for fast computation. The quality of approximation is highly sensitive to the choice of \(\alpha \), and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.