Construction, Analysis, Implementation and Application of New Time Integrators for Large Scale Complex Systems

计算机模拟是科学和工程的重要工具。 依靠计算模型来设计汽车或预测飓风的移动是常见的做法，使科学家和工程师能够显着减少完成这些任务所需的时间和资源。 我们感兴趣的模拟现象的复杂性不断增加。 对复杂系统动力学的预测计算机模型的需求，例如蛋白质分子的折叠或海啸的运动，不断增长，并且不能仅通过计算硬件的进步来满足。 先进的数学方法和计算技术是提高预测计算模型的保真度和效率的一个组成部分。特别是，需要能够模拟复杂系统的时间演化的数值方法来创建可以准确有效地预测系统在长时间内的行为的计算机模型。 拟议的研究重点是建设先进的数值技术，使模拟复杂系统的动态演变在广泛的时间尺度。 该项目将包括这些方法的理论发展以及它们在数值天气预报和气候建模等具体问题上的实施和应用。 这项工作将提供可用于从生物化学和地球工程到流体动力学和等离子体物理学等各种科学和工程领域的软件包。 该项目将从理论和实践的角度推进时间整合的最新技术。越来越清楚的是，快速高阶离散化方法来解决复杂现象的空间和时间尺度是必不可少的，以确保计算模型是足够有效和值得信赖的。虽然大量的研究工作一直致力于发展高阶空间离散方法，如有限元或谱方法，先进的时间积分器的建设和应用的研究滞后于大规模复杂问题。拟议的研究将集中在开发新的时间积分算法，显着提高目前可用的方法的效率。该项目旨在构建，分析和实施时间积分方案，可以同时利用显式，隐式和指数积分方法。重要的理论问题，如这些技术的准确性和收敛性将得到解决。一般的积分器以及类的方法，可以利用特定的问题结构将被导出。此外，将为天气和气候预测问题创建定制设计的时间积分器，以最大限度地提高这些领域常规使用的计算模型的效率。 最有效的时间积分计划，作为本项目的结果开发将实施的一部分，公开提供的软件包设计的串行和并行计算平台。 这项研究的成果也将被整合到应用数学课程的核心研究生课程在加州大学默塞德。

Computer simulations are an essential tool of science and engineering. Relying on computational models to design a car or predict movement of a hurricane is commonplace practice that allows scientists and engineers to significantly reduce time and resources required to accomplish these tasks. The complexity of phenomena we are interested in simulating is constantly increasing. Demand for predictive computer models of complex systems dynamics, such as folding of a protein molecule or movement of a tsunami, is ever-growing and cannot be met with advances in computational hardware alone. Advanced mathematical methods and computational techniques are an integral component of improving fidelity and efficiency of predictive computational models. In particular, numerical methods, which enable simulation of the time evolution of a complex system, are needed to create computer models that can accurately and efficiently predict the behavior of the system over long periods of time. The proposed research focuses on construction of advanced numerical techniques that enable simulating dynamics of complex systems that evolve over a wide spectrum of temporal scales. The project will include both development of theory of such methods as well as their implementation and application to specific problems such as numerical weather prediction and climate modeling. This work will make available software package that can be utilized in a wide variety of scientific and engineering fields ranging from biochemistry and geo-engineering to fluid dynamics and plasma physics. The proposed project will advance the state-of-the-art in time integration from theoretical and practical perspectives. It has become increasingly clear that fast high-order discretization methods to resolve spatial and temporal scales of complex phenomena are essential to ensure that computational models are sufficiently efficient and trustworthy. While extensive research efforts have been dedicated to the development of high-order spatial discretization approaches such as finite-element or spectral methods, research on construction and application of advanced time integrators to large-scale complex problems lags behind. The proposed research will focus on development of new time integration algorithms that significantly improve the efficiency of currently available methods. The project aims to construct, analyze and implement time integration schemes that can simultaneously take advantage of explicit, implicit and exponential integration approaches. Important theoretical questions such as accuracy and convergence of these techniques will be addressed. General integrators as well as classes of methods that can exploit particular problem structure will be derived. In addition, custom-designed time integrators will be created for weather and climate prediction problems to maximize the efficiency of computational models routinely used in these fields. The most efficient time integration schemes developed as a result of this project will be implemented as part of a publicly available software package designed for serial and parallel computational platforms. Outcomes of this research will also be integrated into the core graduate curriculum of the applied mathematics program at UC Merced.