Numerical Methods for Multiple Scale Problems in Wave Propagation: Efficient Approximation of Integral Operators in the Time Domain

开发用于模拟波浪的有效和可靠的计算工具的许多主要障碍植根于普遍存在的多个空间尺度。该项目的重点是详细研究与克服这些障碍有关的选定问题。一个统一的特点，所解决的问题是，所有涉及到有效的近似评估的积分算子在空间和时间。虽然这项工作，如果成功的话，有可能影响许多科学和工程学科，努力将针对航空声学和电磁学的问题。准确地说，将发展以下方面：（i）准确、有效和可靠的计算域截断方法，使直接模拟只在介质复杂或非线性效应重要的区域进行;（ii）有效时间─步进方法，允许简单处理高分辨率或几何细节的集中区域。波传播问题在许多领域具有根本的重要性，应用科学与技术它们涵盖了广泛的物理学（电磁学，流体和固体力学），但共享基本的数学属性。波的定义特征是它能够相对于其基本尺寸（波长）传播很长距离，携带关于它所传播的介质的详细信息。因此，波是我们探索自然和交流的主要方法。这个基本特征的结果是波传播问题通常涉及不同的空间尺度-从散射体的几何细节到传播距离。这些多尺度，反过来，导致计算分析的困难。特别是，它们的统一分辨率将导致令人望而却步的自由度。因此，必须开发的方法，可以集中计算资源只在需要的地方，提供我们考虑的问题的主要动机。除了这项分析，该计划还将与正在构建软件的研究人员合作，这些软件用于模拟喷气噪声、复杂结构中的电磁散射以及通用波传播问题。因此，研究中的任何积极进展都可以尽快投入使用。

Many of the main obstacles to the development of efficient and reliable computational tools for simulating waves are rooted in the multiple spatial scales which are universally present. The focus of this project is the detailed study of select questions which are relevant for overcoming these obstacles. A unifying feature of the questions addressed is that all involve the efficient approximate evaluation of integral operators in space and time. Although the work, if successful, has the potential to impact numerous scientific and engineering disciplines, the efforts will be directed towards problems in aeroacoustics and electromagnetics. Precisely, the following will be developed: (i) Accurate, efficient and reliable computational domain truncation methods, allowing the direct simulations to take place only in regions where the medium is complex or where nonlinear effects are important; (ii) Efficient time-stepping methods allowing the simple treatment of concentrated regions of high resolution or geometric detail.Wave propagation problems are of fundamental importance in many areas of applied science and technology. They encompass a wide range of physics (electromagnetics, fluid and solid mechanics), but share essential mathematical properties. The defining characteristic of a wave is its ability to travel long distances relative to its basic dimension, the wavelength, carrying detailed information about the medium through which it has traveled. For this reason, waves are the primary method by which we probe nature and communicate. A consequence of this fundamental characteristic is that wave propagation problems typically involve disparate spatial scales - from the geometrical details of scatterers through a range of wavelengths to the propagation distances. These multiple scales, in turn, lead to difficulties in computational analysis. In particular, their uniform resolution would lead to a prohibitive number of degrees of freedom. Thus methods must be developed which can concentrate computational resources only where they are needed, providing the primary motivation for the problems we consider. In addition to this analysis, the plan is to collaborate with researchers who are building software for simulating jet noise, electromagnetic scattering in complex structures, as well as general-purpose wave propagation problems. Thus any positive developments from the research can come into use as rapidly as possible.