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

演化偏微分方程的局部指数时差法研究

• 批准号: 1818438
• 负责人: Lili Ju
• 金额: $ 15万
• 依托单位: University of South Carolina at Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1818438&HistoricalAwards=false
• 关键词: Study; Localized; Exponential; Time; Differencing

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.

许多重要的物理现象是通过半线性或完全非线性进化的部分微分方程来建模的。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异方法来提高基于指数积分的方法来解决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的广泛相关应用问题。拟议的工作具有实践意义，具有重大影响，因为开发的方法在现代超级计算机系统上具有高度扩展性，并且可以作为模拟这些严重问题的有效，准确和稳定的计算工具。该项目产生的直接和变革性创新将大大改善许多领域的建模和计算能力，例如设计新材料的设计以及从破裂的油库中回收的油。 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.另一方面，针对许多经典的时间整合方法已经建立了域的分解方法，但没有足够的关注和工作专门用于指数积分器。该项目涉及对基于域分解的迭代和非读力局部指数差异方法的开发和分析的详尽研究，其中一系列时间依赖于时间依赖的标量扩散方程作为原型问题。 PI还将应用开发的方法来研究由材料科学和石油工程产生的多组分和多相系统的某些相场模型。该项目将通过数值研究提供新的见解，以理解合金的宏观特性和可靠性，以及对碳氢化合物的物理现象（例如液滴，气泡和毛细管压力）的物理现象（例如液滴，气泡和毛细管压力）。该奖项反映了NSF的法定任务，并通过评估智能委员会和广泛的范围来评估支持者，并具有值得的支持。

项目成果

Maximum Bound Principles for a Class of Semilinear Parabolic Equations and Exponential Time-Differencing Schemes

一类半线性抛物型方程的最大界原理和指数时差格式

• DOI: 10.1137/19m1243750
• 发表时间: 2021-06-01
• 期刊: SIAM REVIEW
• 影响因子: 10.2
• 作者: Du，Qiang;Ju，Lili;Qiao，Zhonghua
• 通讯作者: Qiao，Zhonghua

Convergence Analysis of Exponential Time Differencing Schemes for the Cahn-Hilliard Equation†

• DOI: 10.4208/cicp.2019.js60.12
• 发表时间: 2019-06
• 期刊: Communications in Computational Physics
• 影响因子: 3.7
• 作者: Xiao Li

Adaptive Exponential Time Integration of the Navier-Stokes Equations

• DOI: 10.2514/6.2020-2033
• 发表时间: 2020-01
• 作者: Shu-Jie Li;L. Ju;H. Si

Exponential Time-Marching method for the Unsteady Navier-Stokes Equations

• DOI: 10.2514/6.2019-0907
• 发表时间: 2019-01
• 期刊: AIAA Scitech 2019 Forum
• 作者: Shu-Jie Li;L. Ju

Localized Exponential Time DifferencingMethod for Shallow Water Equations: Algorithms and Numerical Study

• DOI: 10.4208/cicp.oa-2019-0214
• 发表时间: 2021-06
• 期刊: Communications in Computational Physics
• 影响因子: 3.7
• 作者: X. Meng