Radiation conditions for waves in periodic and stochastic media

• Applicant: Professor Dr. Ben Schweizer
• Subject Area: Mathematics
• Term: from 2019 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 431081410

• Final Report Year: 2024

Final Report Abstract

The project investigated radiation conditions for the Helmholtz equation and the wave equation, the focus was on periodic and stochastic media. If a numerical method is used to solve these equations (approximatively), the unbounded domain must be replaced by a bounded domain. The question is then what boundary conditions should be imposed on the artificially introduced boundaries. We note an important difference to homogeneous materials: It is not desirable to impose a non-reflecting condition, since the heterogeneous medium in the outer area does lead to reflections. In this project, questions in this research field were investigated and partially answered. A first work was concerned in the setting of homogenization limits (i.e., in the case that the periodic structure has a small scale compared to the wavelength); in this case, homogenization theory suggests a boundary condition, which was analyzed in the first publication. A second paper was devoted to the development of a numerical method which is exact in the case of closed waveguides (exact in the sense of the numerical method). We worked on the algorithmic side in order to develop a scheme that is highly efficient in terms of algorithmic complexity. Another work examines stochastic media. Here, we could derive one negative result, namely that stochastic homogenization necessarily fails on large time scales. The more detailed investigation provides a connection between stationary correctors of the problem and the time scale on which the stochastic homogenization homogenization delivers correct results: From the growth of the correctors one can read off the time scales on which the predictions of stochastic homogenization are correct. Finally, regarding radiation conditions for the Helmholtz equation, an existence and uniqueness result was proven with simple methods.

Publications

• Domain truncation methods for the wave equation in a homogenization limit. Applicable Analysis, 101(12), 4149-4170.
  Schäffner, Mathias; Schweizer, Ben & Tjandrawidjaja, Yohanes
  
• A radiation box domain truncation scheme for the wave equation. IMA Journal of Numerical Analysis, 44(2), 920-944.
  Schweizer, B.; Schäffner, M. & Tjandrawidjaja, Y.