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具有材料不连续的介质中的波的传播。潜在的应用包括电磁和声学方面的广泛问题。在数学上，传播是由具有不连续系数的波动方程(无论是在频域还是在时间域)控制的。系数中的不连续通常是第一类。当构造高阶数值近似时，尤其是当它们不与离散化网格对齐时，它们是一个主要的挑战。另一方面，具有高阶离散化对于获得稳健的预测能力是至关重要的，因为它缓解了每波长点数的限制，并且更适合于多尺度问题，例如计算大背景(例如前向传播的激光光束)中的小尺度现象(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？S拟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价曲面参数。后者随后与适当的界面条件相结合，从而产生自洽的公式。Calderon？S算子的一个重要优点是可以用差分势方法高效地计算它们的离散对偶。在这样做的过程中，我们可以使用规则网格而不需要自适应，并且可以得到不规则形状区域的高阶近似。所得结果将对偏微分方程数值方法理论的发展做出重要贡献。在实际应用方面，其结果将是一种高效而稳健的数值方法，用于解决各种应用问题。传播的光、声或无线电波必须通过不同性质的材料之间的界面，这是非常常见的。例子很多，从简单的日常设置，如光的空气和玻璃之间的接口，到雷达和声纳的各种应用，到卫星通信，到等离子聚变设备，等等。界面的存在使在计算机上解决这些问题变得更加困难，材料特性在界面上变化很大。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，在这个项目的过程中，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？S投影的先进数学仪器。该项目的结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.