New Methods for the Simulation and Analysis of Waves

波浪模拟和分析的新方法我们将开发、分析、实施和应用新的方法来解决波浪传播问题。我们的主要重点是数值技术，它可以有效和可靠地利用现代计算机的能力来模拟问题，而这些问题的规模迄今为止妨碍了他们的详细研究。大尺度波的传播问题通常会在一个扩展的空间和/或时间域中给出。在工程应用中，该领域也通常涉及具有复杂形状的物体。这些特征都对数值算法提出了困难的挑战。能量向远场的辐射是大多数波系统的一个特征，并且与大的空间域有关。为了使计算解可行，必须人为地截断域。这种截断引入了难以估计或减少的误差。近年来，我们与其他研究人员一起开发了强制精确截断的新技术，解决了一些特殊但重要的情况下的这个问题。作为这个项目的一部分，我们将扩展这些新方法的适用性。长时间模拟对数值近似的精度也提出了特殊的要求。特别是，波速的小误差或数值阻尼的影响通常会累积而产生大的误差。因此，必须使用高度精确的方法。然而，这些通常难以应用于复杂的几何形状。我们将致力于开发和分析具有高阶精度的新方法，以便更容易地在具有复杂形状的物体附近使用。最后，我们将研究与可压缩Navier-Stokes方程解的存在性、光滑性和渐近逼近性有关的一些基本问题，这些方程描述了最常见的液体和气体的运动。我们特别感兴趣的是与声速相比速度较慢的流。这种流动的典型模型是不可压缩的纳维-斯托克斯方程，它的数学理论虽然还很不完整，但已经得到了更好的发展。然而，我们保留了可压缩效应，并从理论上和模拟上研究了这两个系统解之间的关系。在我们的工作中，我们专注于一些具体的物理系统，主要来自声学和流体动力学领域。因此，我们希望增强我们的能力，使用高性能计算来预测声音与流体流动的产生和相互作用。然而，由于波动理论的普遍重要性及其数学描述的潜在统一性，我们的大多数结果将直接适用于电磁学和弹性等不同领域。从长远来看，提高模拟海浪的能力将有重要的应用，包括减少飞机噪音，改进雷达和声纳成像，地震影响的可预测性，以及设计更好的通信系统。

New Methods for the Simulation and Analysis of Waves Thomas Hagstrom 9971772We will develop, analyze, implement and apply new methods for the solutionof wave propagation problems. Our primary focus is on numerical techniques which can efficiently and reliably utilize the capabilities of modern computers to simulate problems whose scale has hitherto precluded their detailed study. A large scale wave propagation problem will generally be given in an extended spatial and/or temporal domain. In engineering applications, the domain will also usually involve bodies with complex shapes. These features all pose difficult challenges to numerical algorithms. The radiation of energy to the far field is a feature of most wave systems and is associated with large spatial domains. To make computational ssolutions feasible, the domain must be artificially truncated. This truncation introduces errors which have been difficult to estimate or reduce. In recent years, we, along with other researchers, have developed new techniques for imposing accurate truncations, which solve this problem in some special but important cases. As part of this project, we will extend the applicability of these new methods. Long time simulations also make special demands on the accuracy of numerical approximations. In particular, small inaccuracies in the wave speeds or the effects of numerical damping will generally accumulate to produce large errors. Therefore, highly accurate methods must be used. However, these are typically difficult to apply in complex geometries. We will work on the development and analysis of new methods with high orders of accuracy which can be more easily utilized near bodies with complicated shapes. Finally, we will study some basicproblems related to the existence, smoothness, and asymptotic approximability of solutions to the compressible Navier-Stokes equations, which describe the motion of most common liquids and gases. We are particularly interested in flows which are slow in comparison with the speed of sound. Such flows are typically modeled by the incompressible Navier-Stokes equations, whose mathematical theory, though still quite incomplete, is better developed. We, however, retain the compressible effects, and study both theoretically and by simulation the relationship between the solutions of the two systems. In our work we focus on some specific physical systems, primarily from thefields of acoustics and fluid dynamics. As such, we hope to enhance our capability to use high-performance computing to predict the production andinteraction of sound with fluid flows. However, due to the general importance of wave theory and the underlying unity of its mathematical description, most of our results will be directly applicable in diverse areas such as electromagnetism and elasticity. In the long term, an improved capability to simulate waves will have important applications including the reduction of aircraft noise, the improvement of radar and sonar imaging, the predictabilityof the effects of earthquakes, and the design of better communications systems.