The Joy and Pain of Skew Symmetry

Abstract

In this paper, we review recent progress on two related issues. Firstly, the discretisation of partial differential equations of quantum mechanics in a semiclassical regime. Due to the presence of a small parameter, such equations exhibit high oscillations and multiscale behaviour, rendering them difficult to discretise. We describe a methodology, using symmetric Zassenhaus splittings in a free Lie algebra, which allows for their exceedingly fast and accurate numerics. The imperative of preserving the unitarity of the underlying flow takes us to the second theme of this paper, approximation of derivatives by skew-symmetric matrices. Here, we identify a gap in the elementary theory of finite-difference approximations: in the presence of Dirichlet boundary conditions, it is impossible to approximate the derivative to order higher than two on a uniform grid! This motivates the investigation of skew symmetry on non-uniform grids, an endeavour which, although still in its infancy, is already replete with interesting results. We conclude by discussing a number of generalisations and open problems.

Fast Computation of Orthogonal Systems with a Skew‐Symmetric Differentiation Matrix

“…Skew-symmetry mimics the self-adjointness of the first derivative operator in the standard L 2 Hilbert space with either zero Dirichlet or periodic boundary conditions, and it confers a wide range of advantages on a numerical method. We refer to [15,17,18] for a wealth of specific examples: in essence, once a differentiation matrix is skew symmetric, it is often easy to prove stability for linear PDEs, as well as conservation of energy whenever it is mandated by the underlying equation.…”

A Family of Orthogonal Rational Functions and Other Orthogonal Systems with a skew-Hermitian Differentiation Matrix

“…In other words, the differentiation matrix of Φ is skew-symmetric, tridiagonal and irreducible. The virtues of skew symmetry in this context are elaborated in (Hairer & Iserles 2016, Iserles 2016) and (Iserles & Webb 2019b) -essentially, once Φ has this feature, spectral methods based upon it typically allow for a simple proof of numerical stability and for the conservation of energy whenever the latter is warranted by the underlying PDE. The importance of tridiagonality is clear, since tridiagonal matrices lend themselves to simpler and cheaper numerical algebra.…”

Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

“…Skew-symmetry mimics the self-adjointness of the first derivative operator in the standard L 2 Hilbert space with either zero Dirichlet or periodic boundary conditions, and it confers a wide range of advantages on a numerical method. We refer to [15,17,18] for a wealth of specific examples: in essence, once a differentiation matrix is skew symmetric, it is often easy to prove stability for linear PDEs, as well as conservation of energy whenever it is mandated by the underlying equation. The matrix in (1.1) is skew symmetric, yet it is a sobering thought that this second-order approximation of the derivative is as good as it gets: no skew-symmetric finite-difference differentiation matrix on a uniform grid may exceed order 2 [17].…”