Orthogonal systems for time-dependent spectral methods

TL;DR

This paper explores the construction and properties of orthonormal systems with skew-symmetric differentiation matrices for spectral methods, focusing on specific weight functions and their implications for approximation quality and computational efficiency.

Contribution

It introduces a simple construction of orthonormal systems with skew-symmetric differentiation matrices and analyzes their properties for specific weight functions, linking weight properties to boundedness and approximation performance.

Findings

Laguerre and ultraspherical weights exhibit separability enabling fast computation.

Boundedness of matrix powers depends on weight function properties.

Optimal parameter choices improve approximation accuracy.

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 $\mathrm{C}^1(a,b)$ 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 a limited number of powers of that matrix is 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 $x^\alpha \mathrm{e}^{-x}$ 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 {\em 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.