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  • 1Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, La Jolla, CA 92093-0411, USA.

Proceedings. Mathematical, Physical, and Engineering Sciences
|October 16, 2019
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A new numerical method efficiently solves Wiener-Hopf problems. This approach uses spectral accuracy, leveraging far-field solution behavior for diffraction problems.

Keywords:
Riemann–HilbertSommerfeldWiener–Hopfscattering

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Area of Science:

  • Computational mathematics
  • Applied mathematics
  • Numerical analysis

Background:

  • Wiener-Hopf (WH) problems are crucial in various scientific fields, including wave propagation and diffraction.
  • Existing numerical methods may lack efficiency or accuracy for complex WH problems.
  • Riemann-Hilbert problems provide a framework for analyzing and solving WH equations.

Purpose of the Study:

  • To develop a fast and accurate numerical method for solving scalar and matrix Wiener-Hopf problems.
  • To demonstrate the effectiveness of the method on problems generalizing the Sommerfeld diffraction problem.

Main Methods:

  • Formulating Wiener-Hopf problems as Riemann-Hilbert problems on the real line.
  • Developing a numerical approach tailored for these Riemann-Hilbert formulations.
  • Exploiting the far-field behavior of solutions to enhance numerical schemes.

Main Results:

  • The proposed numerical method achieves spectrally accurate results.
  • The approach is validated by solving various scalar and matrix Wiener-Hopf problems.
  • The method demonstrates efficiency and accuracy for diffraction-related problems.

Conclusions:

  • The presented numerical method offers a significant advancement for solving Wiener-Hopf problems.
  • Spectral accuracy is attainable by incorporating far-field solution characteristics.
  • This technique provides a robust tool for analyzing wave diffraction and related phenomena.