Temporal Splitting Methods for Multiscale Problems

在许多实际感兴趣的物理系统中，现象发生在异构介质中，其性质在多个尺度上变化，并且在每个尺度上具有不同的值。例子包括过滤器、膜和地球地下的多物理场过程。模拟这些现象以获得准确预测的标准数值方法需要大量的计算工作。该项目的目标是开发一个统一的框架，以准确有效地模拟复杂的多尺度、时间相关的物理现象，包括多孔介质中出现的流动、输运和机械变形。该项目将通过培养从事多学科研究的新一代计算数学家来支持教育。该项目涉及开发和分析新的时间分裂方法，旨在克服模拟多尺度，时间相关物理现象时出现的挑战。该方法基于非平稳多尺度模型的解分解，并将考虑高度耦合的空间和时间异质性。目标是提供一个将时间分裂算法和空间多尺度分解结合起来的通用框架，并对新算法进行严格的理论分析。该项目的具体目标是：(i)研究非平稳多尺度模型模拟的时间分裂算法；（ii）了解多尺度模型中的时空相互作用；(3)分析时间分裂方法，指导空间分解的选择；（iv）为非线性模型设计新的分割方法；(v)测试和演示提出的方法，以改进工程和地球科学中多尺度、时变物理现象的预测。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In many physical systems of practical interest, phenomena occur in heterogeneous media with properties varying at multiple scales and having disparate values on each scale. Examples include multi-physics processes in filters, membranes, and Earth's subsurface. Standard numerical approaches for simulating these phenomena to obtain accurate predictions require tremendous computational effort. The goal of this project is to develop a unified framework for accurately and efficiently simulating complex multiscale, time dependent physical phenomena that involve flow, transport, and mechanical deformations that arise in porous media. The project will support education by training a new generation of computational mathematicians who work in multidisciplinary research.This project involves the development and analyses of novel temporal splitting methods that are designed to overcome challenges that arise when simulating multiscale, time-dependent physical phenomena. The methods are based on solution decomposition for non-stationary multiscale models and will consider space and time heterogeneities that are highly coupled. The goal is to provide a general framework that combines temporal splitting algorithms and spatial multiscale decompositions with a rigorous theoretical analysis of the new algorithms. The specific objectives of the project are: (i) to study temporal splitting algorithms for simulations of non-stationary multiscale models; (ii) to understand space and time interaction in multiscale models; (iii) to analyze temporal splitting approaches to guide the choice for space decomposition; (iv) to design novel splitting approaches for nonlinear models; and (v) to test and demonstrate proposed approaches for improving predictions of multiscale, time-dependent physical phenomena in engineering and geosciences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.