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，无限域必然被截断，从而产生需要对出射波无反射的虚拟边界。对于控制方程为非线性或变系数的计算科学中的许多重要应用，如何以高精度和高效率正确地建立非反射边界条件仍然是一个重大的挑战。这项研究的目的是基于完全匹配层(PML)方法开发精确的无反射边界条件，可用于产生潜在的无反射吸收边界。在过去的几年里，对于线性化的常系数欧拉方程的PML的构造已经取得了相当大的进展。该工作将解决将PML从常量平均流扩展到非常量平均流的一个关键问题，并导致非线性Euler方程和Navier-Stokes方程的PML的发展。这项研究的一个基本要素是了解由有界流支撑的波的相速度和群速度。所提出的方法是真正的多维的，并且可以扩展到非线性问题。将研究偏微分方程组的数学形式、稳定性和适定性、数值精度和计算优势，特别是色散波的作用。理论和计算结果将与现有方法进行比较。该项目还将为研究生提供培训，在该项目中，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。由于非反射边界在计算科学中的普遍存在，这一研究成果有望对广泛的跨学科科学和工程应用产生广泛而重要的影响，如在湍流和转折、湍流混合、边界层控制、空气声学、水下声学和大气波模拟方面的研究。提出的新方法的使用，如果成功，也将提高现有的计算流体力学程序的质量，无论是在学术界和工业界。

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.