AF: Small: General Linear Multimethods for the Time Integration of Multiscale Multiphysics Problems

科学和工程中的许多领域都依赖于对依赖时间的多尺度多物理系统的计算机模拟。这些领域包括机械和化学工程、航空、天体物理、等离子体物理、气象和海洋学、金融、环境科学和城市建模。根据定义，多尺度问题具有在不同时间或空间尺度上演变的相互作用的组成部分。例如，耦合的地球包括大气(以分钟为时间尺度演变)与海洋(以小时为时间尺度)和与极地冰盖(以天为时间尺度)相互作用。计算机模拟以离散的时间步长跟踪系统的演变--跟踪快速大气变化所需的短时间步长对于模拟缓慢变化的冰来说效率低下。即使在大气层内，局部混合化学反应模型(毫秒)也将与污染物远距离流动模型(数周)相耦合。没有一种单一的时间离散化算法可以有效地对所有过程进行建模。为了提高模拟能力，需要开发新的灵活的时间积分算法，以保持组件模型的准确性和稳定性。本项目将在通用线性方法(GLMS)的框架下，开发一种严格的方法来构造和分析依赖于时间的偏微分方程(PDE)的多速率多方法离散。GLM扩展了传统的积分格式，如Runge-Kutta和线性多步，并具有良好的数学性质，使其非常适合于多方法的构造和研究。新的时间步进算法将具有以下特性：(1)不同的时间步长可以在不同的子域中使用以实现效率；(2)不同的离散化可以应用于模拟不同物理过程的算子；(3)它们将具有高阶次的时间精度，并允许有效的时间步长和误差控制；(4)线性和非线性稳定性约束仅对步长施加局部限制。这些新方法将在高质量的软件中实现，并将应用于大气污染预测中的真实模拟。这一研究成果将推动大规模、多尺度、多物理模拟领域的发展，这将使科学和工程的几个主要领域受益。新型计算高性能计算算法的发展及其在实际问题中的应用为培养博士后和研究生提供了一个极好的机会。PI将广泛传播在这项研究期间开发的算法和软件。

Many fields in science and engineering rely on computer simulations of time-dependent multiscale multiphysics systems. These fields include mechanical and chemical engineering, aeronautics, astrophysics, plasma physics, meteorology and oceanography, finance, environmental sciences, and urban modeling.Multiscale problems, by definition, have interacting components that evolve at different temporal or spatial scales. For example, the coupled Earth includes the atmosphere (evolving at time scales of minutes) interacting with oceans (time scales of hours) and with polar ice caps (time scales of days). Computer simulations track the evolution of systems at discrete time steps -- the short time steps needed to track rapid atmospheric changes are inefficient for modeling the slow-changing ice. Even within the atmosphere models for local mixing chemical reactions (miliseconds) will be coupled to models of long distance flow of pollutants (weeks). No single time discretization algorithm can efficiently model all processes. Improvements in simulation capabilities require the development of new flexible time integration algorithms that preserve the accuracy and stability of the component models.This project will develop a rigorous approach to the construction and analysis of multirate multimethod discretizations for time-dependent partial differential equations (PDEs)in the framework of General Linear Methods (GLMs). GLMs extend traditional integration schemes such as Runge-Kutta and Linear Multistep, and enjoy favorable mathematical properties that make them very well-suited for the construction and study of multimethods. The new time stepping algorithms will enjoy the following properties: (1) different time steps can be used in different subdomains to achieve efficiency; (2) different discretizations can be applied to operators modeling different physical processes; (3) they will have high order of temporal accuracy, and allow for efficient time step and error control; and (4) linear and nonlinear stability constraints impose only local restrictions on the step size. The new methods will be implemented in high quality software, and will be applied to real-life simulations arising in the prediction of atmospheric pollution.The outcomes of this research will advance the field of large, multiscale, multiphysics simulations, which will benefit several major fields in science and engineering. The development of novel computational high performance computing algorithms and their application to real problems provide an excellent opportunity for training postdocs and graduate students. The PI will broadly disseminate the algorithms and software developed during this research.