Advanced Surface Wave Methods for High-Frequency Scattering Problems

In this article, we explore a new approach to solving high-frequency scattering problems in various fields, including electromagnetics, acoustics, and elasticity. These problems are challenging due to the complexity of the governing equations, which lead to numerical instability and slow convergence. To address these issues, we propose the use of "on-surface radiation conditions," a novel technique that leverages surface waves intrinsic to the governing equations.
The key idea behind on-surface radiation conditions is to modify the domain of interest by adding a fictitious surface that supports functions represented on a larger interval. This allows us to reduce the computational complexity of the problem while maintaining accuracy. By combining this technique with fast multipole methods and sweeping algorithms, we can accelerate the numerical solver and achieve better performance in terms of accuracy and efficiency.
To illustrate the effectiveness of our approach, we apply it to several examples from different fields, including electromagnetic scattering by a dielectric sphere and acoustic scattering by a circular cylinder. In each case, we show that on-surface radiation conditions can significantly improve the accuracy and efficiency of the numerical solver.
Our work builds upon previous research in surface wave propagation and the on-surface radiation condition method. However, we introduce a new approach to combining these techniques, which leads to improved performance and flexibility. Our results demonstrate the potential of on-surface radiation conditions for solving high-frequency scattering problems in a wide range of applications.
In summary, on-surface radiation conditions offer a powerful tool for accelerating numerical solvers in high-frequency scattering problems. By leveraging surface waves intrinsic to the governing equations, we can reduce computational complexity while maintaining accuracy. Our approach combines the advantages of fast multipole methods and sweeping algorithms, making it a versatile and effective solution for a variety of applications.