Approximate solutions to Wiener-Hopf equations via the implicit quadrature scheme

WHTW02 - WHT Follow on: the applications, generalisation and implementation of the Wiener-Hopf Method

Obtaining approximate solutions to Wiener-Hopf equations is made difficult by the intricate coupling of near and far field effects. Replacing a term with a numerical approximation that is accurate in a region of the complex plane can have unexpected consequences due to inaccuracies that occur elsewhere. In this presentation, we will discuss the ‘implicit quadrature scheme’, a numerical method that can accurately solve matrix Wiener-Hopf equations.  The scheme does not rely on obtaining an approximate equation that can itself be solved by some exact procedure, but instead solves the Wiener-Hopf equation directly, in a single step.  Unknown functions are represented by Cauchy integrals, and the only approximation occurs when these are eventually evaluated using quadrature formulae.  In this respect, the method is similar to the standard procedure for solving scalar Wiener-Hopf equations, since this typically requires certain functions to be represented as Cauchy integrals, and these must be computed numerically in most cases. The main difference is that the implicit quadrature scheme requires the construction and inversion of a linear system of algebraic equations. The size of this system might once have been considered ‘large’ but on modern computers it can usually be solved in a few seconds at most. The implicit quadrature scheme was introduced in [1], as a means of solving a matrix Wiener-Hopf equation that contains very complicated terms. This work was presented at the INI in 2019.  A numerical library that can be used to easily obtain solutions to Wiener-Hopf equations via the implicit quadrature scheme then began development, funded by an EPSRC Mathematical Sciences small grant (EP/W000504/1). We will discuss the capabilities and limitations of this library, and show results obtained for problems for which accurate solutions are available by other means. [1] Thompson, I. ’’Wave diffraction by a rigid strip in a plate modelled by Mindlin theory.’’  Proceedings of the Royal Society of London A 476 (2243), 2020.

This talk is part of the Isaac Newton Institute Seminar Series series.