Numerical Methods for Wave Propagation Problems: Efficient Resolution of Multiple Scales

我们研究的重点将是详细研究我们认为对开发波传播问题的可靠，高效和通用计算工具至关重要的问题。这些反过来又会对科学和工程的许多领域产生长期影响。准确地说，我们将：（i）进一步发展精确的方法，用于截断需要完全近似的区域附近的计算域，将我们以前构建的成功方法的应用范围扩展到非均匀和各向异性介质以及多尺度计算;（ii）构建和分析新的高分辨率近似方案，从而能够精确模拟每个波长接近最佳的自由度，温和的时间步长稳定性限制，并且易于与网格生成软件耦合，以有效地处理复杂几何形状的问题;（iii）将我们的方法应用于航空声学中的困难问题;（iv）与其他正在构建和维护高质量波浪模拟软件的计算科学家合作。波的传播现象在自然界中普遍存在。虽然波可以由从电流到湍流到大地震的物理过程产生，但它们的基本特征允许统一的数学描述。从现代计算机模拟的角度来看，我们有理由希望能够构建出普遍适用的工具，这些工具将有助于回答整个基础科学和应用科学的重要问题。波的计算分析中的挑战是，几乎所有感兴趣的问题都表现出广泛变化的空间尺度。这是一个基本事实的结果，即波传播的距离相对于其特征尺寸，波长。因此，我们将致力于开发方法，使我们能够避免直接计算波场的沿着其路径的任何地方，只集中计算资源，他们是需要的。除了我们在具有广泛应用的基本技术方面的工作外，我们还计划重点研究与射流产生声音及其传播到环境中有关的问题。我们相信，我们将开展的基础研究可以促进商用和军用飞机更好的声音抑制技术的发展。

The focus of our research will be the detailed study of questions we deem crucial to the development of reliable, efficient, and general computational tools for wave propagation problems. These, in turn, can have long-term impacts on numerous fields in science and engineering. Precisely we will:(i) Further develop accurate methods for truncating the computational domain near regions where full approximations are required, extending the range of application of the successful methods we have previously constructed to inhomogeneous and anisotropic media as well as to multiscale computations; (ii) Construct and analyze novel high-resolution approximation schemes enabling accurate simulations with near-optimal degrees-of-freedom per wavelength, mild time-step stability restrictions, and easy coupling with grid grid generation software to efficiently treat problems in complex geometry; (iii) Apply our methods to difficult problems in aeroacoustics; (iv) Collaborate with other computational scientists who are building andmaintaining high-quality software for simulating waves. Wave propagation phenomena are ubiquitous in nature. Although waves may be produced by physical processes ranging from electric currents to turbulent flows to massive earthquakes, their basic features allow a unified mathematical description. From the perspective of simulations on modern computers, it is reasonable to hope that generally applicable tools can be constructed which will be useful in answering important questions throughout the basic and applied sciences. The challenge in the computational analysis of waves is that almost all problems of interest exhibit widely varying spatial scales. This is a consequence of the fundamental fact that waves propagate long distances relative to their characteristic dimension, the wavelength. We thus will work to develop methods which allow us to avoid the direct computation of the wave field everywhere along its path, concentrating computational resources only where they are needed. In addition to our work on basic techniques with broad applications, we plan focused studies on problems related to the generation of sound by jets and its propagation into the environment. We believe that the fundamental studies we will carry out can motivate the development of better sound suppression technologies for commercial and military aircraft.