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This study introduces a fast numerical method for evaluating complex integrals common in physics and mathematics. The new scheme accurately describes wave scattering phenomena, including non-classical diffraction effects.

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Area of Science:

  • Physics
  • Mathematics
  • Wave Scattering
  • Numerical Analysis

Background:

  • Many physical and mathematical phenomena involve integrals with oscillating integrands and multiple critical points.
  • Evaluating these complex integrals, especially those with coalescing criticalities, poses a significant computational challenge.

Purpose of the Study:

  • To develop a fast and efficient numerical scheme for evaluating integrals with oscillating integrands featuring coalescing criticalities.
  • To apply and validate the proposed scheme in the context of wave scattering problems.

Main Methods:

  • A novel numerical scheme based on the regularized composite Simpson's rule is proposed.
  • The method is demonstrated by analyzing the scattering of an elastic plane wave by a stress-free half-plane crack.

Main Results:

  • The proposed scheme efficiently evaluates challenging integrals with branch points, stationary phase points, and poles.
  • It accurately describes non-classical diffraction effects near critical rays, including far-field spikes and near-field interference ripples.

Conclusions:

  • The developed numerical scheme provides an effective solution for evaluating complex oscillatory integrals.
  • This advancement aids in understanding intricate wave phenomena and diffraction effects in various scientific fields.