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

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介质中的传播。潜在的应用包括电磁学和声学的广泛问题。数学上，传播是由具有不连续系数的波状方程（在频域或在时域）控制的。系数的不连续是典型的第一类。它们在构建高阶数值近似时提出了一个主要挑战，特别是当它们不与离散网格对齐时。另一方面，具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的限制，并且也更适合于多尺度问题，例如在大背景（例如前向传播激光束）下计算小尺度现象（例如光学中的非线性后向散射）。在项目的过程中，我们将在Calderon的帮助下解决上述问题。S伪微分边界投影。这些算子允许我们得到解的等效表面参数化。后者随后与适当的界面条件相结合，从而产生自洽公式。卡尔德隆的关键优势是什么？S算子的一个重要特点是它们的离散对应算子可以用差分势的方法有效地计算出来。在这样做的时候，人们可以使用规则网格而不进行自适应，并获得不规则形状域的高阶近似。预期的结果将对偏微分方程数值方法理论作出重要贡献。在实际方面，结果将是一个有效和稳健的数值方法来解决各种应用问题。传播的光、声或无线电波必须通过具有不同性质的材料之间的界面，这是很常见的。例子很多，范围从简单的日常设置，如空气和玻璃之间的光接口，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的特性变化很大，这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，相应的公式有许多重要的组成部分，在项目的过程中，我们将开发和测试一种通用的数值方法来解决各种这样的问题。这种方法将利用被称为卡尔德隆？年代的预测。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，并将对声学，电磁学和光学的应用具有重要意义。

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.