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T-systems: a theory of orthonormal functions with a tridiagonal differentiation matrix

Arieh Iserles, Marcus Webb

Abstract

The starting point of this paper is that a spectral method is essentially a combination of an orthonormal basis of the underlying Hilbert space with Galerkin conditions. The choice of an orthonormal basis depends on a number of desirable features which we explore in the context of spectral methods for time-dependent partial differential equations in a single space dimension. A central role in ensuring many of the above features is played by the differentiation matrix of the underlying orthonormal system. In particular, it is beneficial if this matrix is skew-symmetric and tridiagonal. While orthonormal systems with this feature have been characterised in A. Iserles & M. Webb, ``Orthogonal systems with a skew-symmetric differentiation matrix'', Found. Comput. Maths, 19 (2019), 1191--1221, employing Fourier transforms, in this paper we provide an alternative characterisation using the differential Lanczos algorithm, which can be implemented constructively. It is valid for inner products that obey an `integration-by-parts condition', inclusive of $L_2$ and Sobolev norms on the real line. Motivated by quest for integration methods that conserve Hamiltonian energy, we conclude the paper replacing inner products by more general sesquilinear forms and presenting preliminary results. Here the Fourier transform characterisation generalises to spectral theory of Schrödinger operators and the differential Lanczos algorithm generalises to the differential Arnoldi algorithm.

T-systems: a theory of orthonormal functions with a tridiagonal differentiation matrix

Abstract

The starting point of this paper is that a spectral method is essentially a combination of an orthonormal basis of the underlying Hilbert space with Galerkin conditions. The choice of an orthonormal basis depends on a number of desirable features which we explore in the context of spectral methods for time-dependent partial differential equations in a single space dimension. A central role in ensuring many of the above features is played by the differentiation matrix of the underlying orthonormal system. In particular, it is beneficial if this matrix is skew-symmetric and tridiagonal. While orthonormal systems with this feature have been characterised in A. Iserles & M. Webb, ``Orthogonal systems with a skew-symmetric differentiation matrix'', Found. Comput. Maths, 19 (2019), 1191--1221, employing Fourier transforms, in this paper we provide an alternative characterisation using the differential Lanczos algorithm, which can be implemented constructively. It is valid for inner products that obey an `integration-by-parts condition', inclusive of and Sobolev norms on the real line. Motivated by quest for integration methods that conserve Hamiltonian energy, we conclude the paper replacing inner products by more general sesquilinear forms and presenting preliminary results. Here the Fourier transform characterisation generalises to spectral theory of Schrödinger operators and the differential Lanczos algorithm generalises to the differential Arnoldi algorithm.
Paper Structure (31 sections, 6 theorems, 104 equations, 8 figures, 2 algorithms)

This paper contains 31 sections, 6 theorems, 104 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Spectral methods for time-dependent PDEs
  3. Differentiation matrices
  4. On differentiation matrices
  5. Skew-Hermitian differentiation matrices
  6. T-systems
  7. The Cauchy problem via the Fourier transform
  8. Periodic boundary conditions via the Fourier transform
  9. The theory
  10. Example: Simple periodic setting
  11. Example: The Charlier measure
  12. Twisted periodic boundary conditions
  13. Zero Dirichlet boundary conditions
  14. The differential Lanczos algorithm
  15. From $\varphi_0$ to $\Phi$
  16. ...and 16 more sections

Key Result

Proposition 4

It is true that where each $q_n$ is an $n$th-degree monic polynomial that obeys the three-term recurrence relation

Figures (8)

  • Figure 2.1: The functions $\varphi_n$, $n=0,1,2,3$, for the first periodic example and $a=\frac{1}{2}$.
  • Figure 2.2: The functions $\varphi_n$, $n=0,1,2,3$, for the Charlier measure and $a=\frac{1}{2}$.
  • Figure 2.3: The functions $\varphi_n$, $n=0,\ldots,5$, for the $\varphi_0$ given by (\ref{['eq:2.17']}).
  • Figure 3.1: The commuting triangle of mappings between the set of seeds $\mathcal{S}$, the set of determinate Borel measures $\mathcal{B}$ and the set of non-separable Jacobi matrices $\hbox{J}$.
  • Figure 3.2: $b_n$ for $n=0,\ldots,10$ and the seed (\ref{['eq:3.4']}).
  • ...and 3 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Theorem 5
  • Definition 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10