Study of Dispersive Waves and Development of Accurate Nonreflecting Boundary Conditions

在涉及开放物理边界的数值模拟中，由于有限计算资源的限制，无限域必须被截断，从而创建需要对传出波无反射的虚拟边界。在计算科学中，当控制方程为非线性或具有非常系数时，如何高精度、高效率地构造无反射边界条件是一个重大的挑战。建议的研究旨在开发精确的无反射边界条件的基础上完美匹配层（PML）的方法，可用于产生潜在的无反射吸收边界。在过去的几年中，常系数线性化Euler方程的PML构造取得了很大的进展。本文的工作解决了将PML从定常平均流扩展到非定常平均流的关键问题，并为非线性Euler方程和Navier-Stokes方程的PML的发展奠定了基础。在这项研究中的一个基本要素是有界流支持的波的相速度和群速度的理解。所提出的方法是真正的多维和可扩展的非线性问题。将研究关于偏微分方程的数学公式、稳定性和适定性、所提出的方法的数值精度和计算优势，特别是色散波的作用的问题。理论和计算结果将与现有的方法进行比较。该项目还将为研究生提供培训，其中一个更传统的应用数学主题，色散波分析，与核心科学计算问题密切相关。由于无反射边界在计算科学中的普遍性，本研究的结果预计将对广泛的跨学科科学和工程应用产生广泛而重大的影响，例如湍流和转捩，湍流混合，边界层控制，航空声学，水下声学和大气波建模的研究。使用所提出的新方法，如果成功的话，也将提高现有的计算流体动力学代码在学术界和工业界的质量。

In numerical simulations that involve an open physical boundary, the infinite domain is necessarily truncated due to the limitation of finite computational resources, creating virtual boundaries thatneed to be reflectionless to out-going waves. It remains a significant challenge to formulate correctly the nonreflecting boundary condition with high accuracy and efficiency for many important applicationsin computational sciences where the governing equations are nonlinear or have non-constant coefficients. The proposed research is aimed at developing accurate nonreflecting boundary conditions based on thePerfectly Matched Layer (PML) methodology that can be used to produce potentially reflectionless absorbing boundaries. In the past few years, considerable progresses have been made in constructing PML for thelinearized Euler equations with constant coefficients. The proposed work will resolve a key issue in extending PML from constant to non-constant mean flows and lead to the development of PML for nonlinear Euler and Navier-Stokes equations. An essential element in this research is the understanding of phase and group velocities of waves supported by a bounded flow. The proposed approach is genuinely multi-dimensional and extendible to nonlinear problems. Issues about mathematical formulation, stability and well-posedness of the partial differential equations, numerical accuracy and computational advantages of the proposed method, in particular the role of dispersive waves, will be studied. Theoretical as well as computational results will be compared with existing methods. This project will also provide training of graduate students in which a more traditional applied math topic, dispersive wave analysis, is closely coupled with a core scientific computing issue. Due to ubiquitousness of nonreflecting boundaries in computational sciences, the findings of this research are expected to have a broad and significant impact on a wide range of interdisciplinary scientific and engineering applications, such as those in the study of turbulence and transition, turbulent mixing, boundary layer control, aeroacoustics, underwater acoustics and atmospheric wave modeling. The use of the proposed new method, when successful, will also improve the quality of existing computational fluid dynamics codes both in the academia and industry.