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

我们的研究重点将是对我们认为对开发可靠、高效和通用的波传播问题计算工具至关重要的问题的详细研究。这些反过来又会对科学和工程领域的许多领域产生长期影响。确切地说，我们将：(I)进一步发展精确的方法，在需要完全近似的区域附近截断计算区域，将我们先前构造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)构造和分析新的高分辨率近似格式，使得能够以接近最佳的每波长自由度、温和的时间步长稳定性限制以及容易与网格生成软件的耦合来精确模拟，以有效地处理复杂几何形状中的问题；(Iii)将我们的方法应用于气动声学中的困难问题；(Iv)与正在开发和维护高质量的海浪模拟软件的其他计算科学家合作。波的传播现象在自然界中是普遍存在的。虽然波可能是由从电流到湍流再到大地震的物理过程产生的，但它们的基本特征允许进行统一的数学描述。从现代计算机模拟的角度来看，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中的重要问题。波的计算分析中的挑战在于，几乎所有感兴趣的问题都表现出大不相同的空间尺度。这是一个基本事实的结果，即波相对于其特征维度波长传播很长的距离。因此，我们将致力于开发方法，使我们能够避免在其路径上的任何地方直接计算波场，只将计算资源集中在需要的地方。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喷气式飞机产生的声音及其传播到环境中有关的问题。我们相信，我们将开展的基础性研究可以推动商用和军用飞机更好的抑声技术的发展。

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.