Multiscale Computation of Highly Oscillatory Dynamical Systems

高振荡动力系统的多尺度计算

• 批准号: 1217203
• 负责人: Bjorn Engquist
• 金额: $ 49.72万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1217203&HistoricalAwards=false
• 关键词: Multiscale; Computation; Highly; Oscillatory; Dynamical

Highly oscillatory dynamical systems are computationally very challenging. Traditional numerical techniques require several function evaluations per wavelength. This is typically prohibitingly computationally costly. In earlier work the investigators have developed and analyzed numerical algorithms for efficient solution of such systems with well-defined scale separation. In this proposal the investigators tackle the harder problems where the scale separation is less clear and where the decomposition into slow and fast variables is not known. Abstractly, a full-scale model with state variables u is given. A number of slow variables, U, which include the local averages of u(t), is to be computed using a sequence of short time histories of u(t), starting from appropriate initial conditions consistently defined by U(t). Hence, the essential objectives that form this proposal are: (1) Under what conditions and in what sense do the multiscale approaches converge? What is the accuracy in the approximation of U, and what are the stability properties of the methods? (2) How can we find higher order accurate algorithms in a systematic way, when the dynamical system's right hand side can be decomposed into stiff and non-stiff parts? (3) Design such methods with a substantial reduction of computational complexity compared with existing techniques. The proposed research will delineate mathematically what it means for a dynamical system that appear to be highly oscillatory to possess slow modes. With this information, new efficient numerical algorithms can be devised and tested rigorously. The proposed research will establish a mathematical link between a variety of averaging theories and new effective numerical integrators and filtering techniques.The proposed research is at the very mathematical and computational heart of many important applications in science and engineering, from astrophysics to quantum mechanics. In these applications there are rapid oscillations as, for example, vibrations of atoms that affects the overall system on a much slower time scale. This is traditionally difficult to simulate with reasonable computational cost. The theories and algorithms developed in this proposal make such simulations practical and directly relate to important atomistic simulations that are used in place of actual physical experiments. They will have direct consequences, for example, in molecular biology for understanding drug functions at the cellular level, and also in the study of material properties and unstable events such as the formation and propagation of cracks, which may affect electronic components as well as the structural safety of airplanes.

高度振荡的动力系统在计算上是非常具有挑战性的。传统的数值方法需要对每个波长进行多次函数评估。这通常是令人望而却步的计算成本。在早期的工作中，研究人员已经开发和分析了有效解这类具有明确尺度分离的系统的数值算法。在这项建议中，研究人员解决了较困难的问题，即尺度分离不太清楚，以及分解成慢变量和快变量的情况未知。抽象地给出了一个状态变量为u的全尺度模型。从U(T)始终如一地定义的适当初始条件开始，使用u(T)的短时间历史序列来计算包括u(T)的局部平均值的多个慢变量U。因此，形成这一提议的基本目标是：(1)多尺度方法在什么条件下以及在什么意义上收敛？当动力系统的右端可以分解为刚性部分和非刚性部分时，如何才能系统地找到高阶精度算法？(3)与现有技术相比，设计这种方法的计算复杂度有很大的降低。这项拟议的研究将从数学上描绘出，对于一个看起来高度振荡的动力系统来说，拥有慢模式意味着什么。有了这些信息，就可以设计出新的有效的数值算法，并进行严格的测试。这项拟议的研究将在各种平均理论和新的有效的数值积分器和滤波技术之间建立数学联系。拟议的研究处于从天体物理到量子力学的许多重要科学和工程应用的数学和计算核心。在这些应用中，存在快速的振荡，例如，在较慢的时间尺度上影响整个系统的原子振动。这在传统上很难用合理的计算成本进行模拟。该方案中开发的理论和算法使这种模拟变得实用，并与重要的原子模拟直接相关，这些模拟用于代替实际的物理实验。例如，它们将对在细胞水平上了解药物功能的分子生物学产生直接影响，还将在材料特性和不稳定事件的研究中产生直接影响，例如可能影响电子元件和飞机结构安全的裂纹的形成和扩展。