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

小行星9600146 该项目的主要重点是开发，分析和 实施新的、更有效的数值模拟方法 在多个时间和空间尺度的存在下的波传播。 这包括构建有效的离散化方法， 求解线性双曲型方程组的精确辐射条件， 发展自适应谱方法，用于需要高度 精确解，并分析分裂计划的有效 多时标偏微分积分问题 方程要考虑的具体物理应用包括声学 和电磁波的传播， 利用详细模型模拟动态燃烧现象 物理学的。非线性偏微分方程调节系统的基本分析 微分方程组也将进行。 多尺度问题是困难的，因为统一的分辨率 目前最小的尺度将导致一个巨大的计算 问题，即使是最快的机器可能会在未来可用 几十年它们也是科学和 在许多不同的物理现象中， 设置.它们有时是由于需要解决大的问题而产生的 空间域或长时间段，以及其他时间， 出现非常局部化的解决方案功能，如尖锐的波前。在 后一种情况下，它们通常与奇异摄动有关， 控制方程奇异摄动可以被认为是很小的 方程中的变化可能导致大量的定性和定量 解决方案的变化。这类问题也常常可以用非数值的渐近方法来分析，在这项工作中， 以多种方式用于指导和补充数字 计算。将理论进步转化为有用的工具， 考虑更复杂和更全面的数学 强调模型，以最大限度地提高研究对其他 科学和工程学科。

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.