Radiation conditions for waves in periodic and stochastic media

本项目分析了亥姆霍兹方程和波动方程的新辐射条件，我们考虑了周期介质和随机介质。我们强调，在异质介质中，人工边界不应该是非反射边界，因为异质介质在外部域会产生反射。对于亥姆霍兹方程，我们想改进最近提出的一种方案。该方案要求在所谓的辐射盒中，解决方案是输出布洛赫波的线性组合。该格式可以转化为波动方程，我们可以得到一个实用的数值格式-至少在均匀化极限下。利用均匀化理论中的一阶校正来改进数值格式是本课题的一个重要思想。我们在项目的第三部分分析随机介质。我们使用最近引入的泰勒-布洛赫波将上述辐射箱的思想转移到随机设置中。我们的目的是对大波长的新方法进行分析，并对中等波长的新方法进行数值测试。

The project analyzes new radiation conditions for the Helmholtz equation and for the wave equation, we consider periodic and stochastic media. We emphasize that, in heterogeneous media, an artificial boundary should not be a non-reflecting boundary, since the heterogeneous medium in the outer domain creates reflections. For the Helmholtz equation, we want to improve a recently suggested scheme. The scheme demands that the solution is, in so called radiation boxes, a linear combination of outgoing Bloch-waves. The scheme can be transferred to the wave equation, where we can obtain a practical numerical scheme - at least in the homogenization limit. It is a crucial idea of this project to use first order correctors from homogenization theory to improve the numerical scheme. We analyze stochastic media in a third part of the project. We use the recently introduced Taylor-Bloch waves to transfer the above ideas of radiation boxes to the stochastic setting. Our aim is to analyze the new method for large wave-lengths and to test the new method numerically for moderate wave-lengths.