Marcus Webb

Alex Townsend, Marcus Webb, Sheehan Olver

Many standard conversion matrices between coefficients in classical orthogonal polynomial expansions can be decomposed using diagonally-scaled Hadamard products involving Toeplitz and Hankel matrices. This allows us to derive $\smash{\mathcal{O}(N(\log N)^2)}$ algorithms, based on the fast Fourier transform, for converting coefficients of a degree $N$ polynomial in one polynomial basis to coefficients in another. Numerical results show that this approach is competitive with state-of-the-art techniques, requires no precomputational cost, can be implemented in a handful of lines of code, and is easily adapted to extended precision arithmetic.

Marcus Webb, Georg Maierhofer

We introduce an efficient stable algorithm for transforms associated with expansions in Hermite functions interpolated at Hermite polynomial roots. The Hermite transform matrix can be factorised into a diagonal component and an orthogonal matrix, leading to a form which allows both the forward and inverse Hermite transforms to be computed stably. Our novel algorithm computes this factorisation based on the eigendecomposition of the Jacobi matrix associated with Hermite functions. Through numerical experiments, we demonstrate the stability and efficiency gains of this novel method over prior work. Numerical experiments show that the new approach matches or improves on the accuracy of existing stabilized methods, is substantially faster in practice, and enables reliable use of large Hermite expansions in downstream PDE computations. We also provide an open-source implementation, together with reference implementations of previous methods, to facilitate adoption by the community.

Arieh Iserles, Marcus Webb

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.

Philipp Bader, David I McLaren, G. R. W. Quispel et al.

It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge-Kutta method will respect this property for such systems, but it has been shown that no B-Series method can be volume preserving for all volume preserving vector fields (BIT 47 (2007) 351-378 and IMA J. Numer. Anal. 27 (2007) 381-405). In this paper we show that despite this result, symplectic Runge-Kutta methods can be volume preserving for a much larger class of vector fields than Hamiltonian systems, and discuss how some Runge-Kutta methods can preserve a modified measure exactly.