A nonuniform mesh method for wave scattered by periodic surfaces

TL;DR

This paper introduces a novel nonuniform mesh numerical method for simulating acoustic wave scattering by two-dimensional periodic structures with non-periodic incident fields, emphasizing higher-dimensional extension and exponential convergence.

Contribution

The paper develops a new nonuniform mesh method using Gaussian quadrature for efficient, high-accuracy simulation of wave scattering in periodic structures, adaptable to higher dimensions.

Findings

Method achieves exponential convergence.

Numerical experiments confirm theoretical results.

Applicable to higher-dimensional problems.

Abstract

In this paper, we propose a new nonuniform mesh method to simulate acoustic scattering problems in two dimensional periodic structures with non-periodic incident fields numerically. As existing methods are difficult to extend to higher dimensions, we have designed the new method with such extensions in mind. With the help of the Floquet-Bloch transform, the solution to the original scattering problem is written as an integral of a family of quasi-periodic problems. These are defined in bounded domains for each value of the Floquet parameter which varies in a bounded interval. The key step in our method is the numerical approximation of the integral by a quadrature rule adapted to the regularity of the family of quasi-periodic solutions. We design a nonuniform mesh with a Gaussian quadrature rule applied on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results.