# Numerical solution of scattering problems using a Riemann--Hilbert formulation

## TL;DR

This paper introduces a fast, spectrally accurate numerical method for solving scalar and matrix Wiener--Hopf problems by formulating them as Riemann--Hilbert problems on the real line, applicable to diffraction problems.

## Contribution

It develops a novel numerical approach leveraging Riemann--Hilbert formulation for Wiener--Hopf problems, enabling high-accuracy solutions for generalized diffraction scenarios.

## Key findings

- Achieved spectrally accurate numerical solutions.
- Successfully solved generalized Wiener--Hopf problems.
- Demonstrated applicability to classical diffraction problems.

## Abstract

A fast and accurate numerical method for the solution of scalar and matrix Wiener--Hopf problems is presented. The Wiener--Hopf problems are formulated as Riemann--Hilbert problems on the real line, and a numerical approach developed for these problems is used. It is shown that the known far-field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results. A number of scalar and matrix Wiener--Hopf problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the approach.