Study on Localized Exponential Time Differencing Methods for Evolution Partial Differential Equations

许多重要的物理现象都可以用半线性或完全非线性的发展偏微分方程来模拟。该项目的总体目标是通过设计和分析高度可扩展的局部指数时间差分方法来提高基于指数积分器的方法求解这些方程的效率和可扩展性，并将其应用于数值模拟和研究科学和工程中广泛的相关应用问题。所提出的工作是具有重大影响的实际利益，因为开发的方法是高度可扩展的现代超级计算机系统，并可以作为一个高效，准确和稳定的计算工具，这些刚性问题的模拟。该项目产生的直接和变革性创新将大大提高许多领域的建模和计算能力，例如新材料的设计和裂缝油藏的采油。此外，本发明还提供了一种方法，该项目还将为对计算和应用数学感兴趣的研究生提供一个独特的教育机会，让他们参与跨学科的研究环境。由于快速傅里叶变换或Krylov子空间需要密集的数据通信，全局指数时间差分方法的直接并行化通常很难在大规模分布式系统上扩展-基于矩阵指数和向量的乘积的计算。另一方面，区域分解方法已被许多经典的时间积分方法很好地建立，但没有足够的关注和工作一直致力于指数积分。本项目以一类含时标量扩散方程为原型问题，对基于区域分解的迭代和非迭代局部化指数时间差分方法的发展和分析进行了深入的研究。PI还将应用所开发的方法来研究材料科学和石油工程中出现的多组分和多相系统的相场模型。该项目将通过数值研究为理解合金的宏观性质和可靠性以及碳氢化合物流体的物理现象（如液滴、气泡和毛细管压力）提供新的见解。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many important physical phenomena are modeled by semilinear or fully nonlinear evolution partial differential equations. The overall goal of the project is to enhance the efficiency and scalability of exponential integrator-based methods for solving these equations by designing and analyzing highly scalable localized exponential time differencing methods and to apply them to numerically simulate and investigate a wide range of related application problems in science and engineering. The proposed work is of practical interest with significant influences as the developed methods are highly scalable on modern supercomputer systems, and can serve as an efficient, accurate and stable computational tool for simulations of these stiff problems. Direct and transformative innovations resulting from the project will greatly improve modeling and computational capabilities for many fields, such as design of new materials and oil recovery from fractured oil reservoirs. In addition, this project will also offer a unique educational opportunity for graduate students with interests in computational and applied mathematics by having them participate in an interdisciplinary research environment.Direct parallelization of global exponential time differencing methods is often very hard to be scalable on massively distributed systems due to the intensive data communications needed by fast Fourier transform or by Krylov subspace-based calculations for products of matrix exponentials and vectors. On the other hand, domain decomposition approaches have been well established for many classic time integration methods, but not enough attention and work have been devoted to exponential integrators. This project involves a thorough study on the development and analysis of iterative and noniterative localized exponential time differencing methods based on domain decomposition, with a family of time-dependent scalar diffusion equations as the prototype problem. The PI will also apply the developed methods to study some phase field models for multi-component and multi-phase systems arising from materials science and petroleum engineering. This project would offer new insights through numerical investigations to the understanding of the macroscopic properties and reliability of alloys and the physical phenomena (such as liquid droplets, gas bubbles, and capillary pressure) of hydrocarbon fluids.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.