Mathematical Sciences: Computational Analysis of Multiple Scales Problems in Wave Propagation

9600146 Hagstrom这个项目的主要重点是开发、分析和实施新的、更有效的数值方法，以模拟存在多个时间和空间尺度的波的传播。这包括为解线性双曲组构造有效的离散化方法和精确的辐射条件，为需要高精度解的问题开发自适应谱方法，以及为偏微分方程组的多时间尺度问题的有效积分分析分裂格式。要考虑的具体物理应用包括声波和电磁波的传播，以及利用物理的详细模型来模拟动态燃烧现象的挑战性问题。还将对非线性偏微分方程组的控制系统进行基本分析。多尺度问题是困难的，因为存在的最小尺度的统一分辨率将导致令人望而却步的巨大计算问题，即使对于可能在未来几十年中可用的最快的机器也是如此。它们也是许多不同物理环境中具有重要科学和技术意义的现象的共同特征。它们有时是因为需要解决大空间域或长时间段上的问题，有时是因为出现了非常局部化的解特征，如尖锐的波前。在后一种情况下，它们通常与控制方程的奇异摄动有关。奇异摄动可以被认为是方程中的微小变化，它可以导致解的大的定性和定量的变化。这类问题通常也可以用非数值渐近技术来分析，在这项工作中，渐近分析被用在许多方法上来指导和补充数值计算。强调将理论进步转化为有用的工具，并考虑更复杂和更全面的数学模型，以最大限度地发挥研究对其他科学和工程学科的影响。

9600146 Hagstrom This project's primary focus is on the development, analysis and implementation of new and more efficient numerical methods for simulating wave propagation in the presence of multiple temporal and spatial scales. This includes the construction of efficient discretization methods and accurate radiation conditions for solving linear hyperbolic systems, the development of adaptive spectral methods for problems requiring highly accurate solutions, and the analysis of splitting schemes for the efficient integration of multiple time scales problems for partial differential equations. Specific physical applications to be considered include acoustic and electromagnetic wave propagation and the challenging problem of simulating dynamic combustion phenomena making use of detailed models of the physics. Basic analyses of the governing systems of nonlinear partial differential equations will also be undertaken. Multiple scales problems are difficult because the uniform resolution of the smallest scales present would lead to a prohibitively large computational problem, even for the fastest machines likely to be available in the coming decades. They are also a common feature of scientifically and technologically important phenomena in a number of distinct physical settings. They arise sometimes from the need to solve problems on large spatial domains or over long time periods, and other times from the appearance of very localized solution features, such as sharp wavefronts. In the latter case they are generally associated with singular perturbations of the governing equations. Singular perturbations may be thought of as small changes in the equations which can lead to large qualitative and quantitative changes in the solutions. Such problems can also often be analyzed by non- numerical asymptotic techniques and in this work asymptotic analysis is used in a number of ways to guide and complement numerical computations. The translation of theoretical advances into useful tools, and the consideration of more complicated and comprehensive mathematical models is emphasized so as to maximize the impact of the research on other science and engineering disciplines.