High-Order Numerical Solution of Wave-Type Equations with Discontinuous Coefficients

具有不连续系数的波动型方程的高阶数值解

• 批准号: 0810963
• 负责人: Semyon Tsynkov
• 金额: $ 19.98万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-15 至 2011-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0810963&HistoricalAwards=false
• 关键词: Order; Numerical; Solution; Wave; Type

The key objective of the project is to build an efficient numerical method for computing the propagation of waves in the media with material discontinuities. Potential applications include a broad range of problems in both electromagnetism and acoustics. Mathematically, the propagation is governed by wave-like equations (either in the frequency domain or in the time domain) with discontinuous coefficients. Discontinuities in the coefficients are typically of the first kind. They present a major challenge when constructing a high-order numerical approximation, especially when they are not aligned with the discretization grid. Having a high-order discretization, on the other hand, is crucial for obtaining a robust predictive capability, because it alleviates the points-per-wavelength constraint and is also far better suited for multiscale problems, such as computing a small scale phenomenon (e.g., nonlinear backscattering in optics) at a large background (such as the forward propagating laser beam). In the course of the project, we will address the foregoing problem with the help of Calderon?s pseudodifferential boundary projections. These operators allow one to obtain equivalent surface parameterizations of solutions. The latter are subsequently combined with the appropriate interface conditions, which yields a self-consistent formulation. A key advantage of Calderon?s operators is that their discrete counterparts can be efficiently computed using the method of difference potentials. In doing so, one can use regular grids with no adaptation, and obtain high-order approximations for the domains of irregular shape. The anticipated results will make an important contribution to the theory of numerical methods for partial differential equations. On the practical side, the outcome will be an efficient and robust numerical methodology for solving a variety of applied problems.It is very common that the propagating light, or sound, or radio waves have to pass through the interfaces between the materials with different properties. Examples are abundant and range from simple everyday setups, such as the interface between air and glass for light, to various applications of radars and sonars, to satellite communications, to plasma fusion devices, and others. The presence of interfaces, across which the material characteristics vary sharply, makes it more difficult to solve these problems on the computer. However, from the standpoint of mathematics, the corresponding formulations share a number of important components, and in the course of the project we are going to develop and test a universal numerical methodology for solving a variety of such problems. The methodology will exploit the advanced mathematical apparatus known as Calderon?s projections. The results of the project will contribute to both the theory and practice of solving scientific problems on the computer, and will be important for applications in acoustics, electromagnetism, and optics.

该项目的主要目标是建立一种有效的数值方法来计算波在具有材料不连续性的介质中的传播。潜在的应用包括电磁学和声学方面的广泛问题。从数学上讲，传播由具有不连续系数的波状方程（频域或时域）控制。系数的不连续性通常属于第一类。在构造高阶数值近似时，特别是当它们不与离散化网格对齐时，它们提出了重大挑战。另一方面，拥有高阶离散化对于获得强大的预测能力至关重要，因为它减轻了每波长点数的约束，并且也更适合多尺度问题，例如在大背景（例如前向传播的激光束）下计算小尺度现象（例如光学中的非线性后向散射）。在项目过程中，我们将借助卡尔德隆的伪微分边界投影来解决上述问题。这些算子允许人们获得解的等效表面参数化。随后将后者与适当的界面条件相结合，从而产生自洽的配方。 Calderon 算子的一个关键优点是可以使用差势方法有效地计算其离散对应项。这样做时，我们可以使用不进行任何调整的规则网格，并获得不规则形状域的高阶近似。预期结果将对偏微分方程数值方法理论做出重要贡献。在实践方面，结果将是一种有效且稳健的数值方法，用于解决各种应用问题。传播的光、声音或无线电波必须穿过具有不同属性的材料之间的界面是很常见的。例子很丰富，范围从简单的日常设置（例如空气和玻璃之间的光界面）到雷达和声纳的各种应用，到卫星通信，到等离子体聚变装置等等。界面的存在使得材料特性变化很大，使得在计算机上解决这些问题变得更加困难。然而，从数学的角度来看，相应的公式共享许多重要的组成部分，在该项目的过程中，我们将开发和测试一种通用的数值方法来解决各种此类问题。该方法将利用被称为卡尔德隆预测的先进数学工具。该项目的成果将有助于在计算机上解决科学问题的理论和实践，并且对于声学、电磁学和光学领域的应用具有重要意义。