Collaborative Research: Efficient, Stable and Accurate Numerical Algorithms for a class of Gradient Flow Systems and their Applications

合作研究：一类梯度流系统高效、稳定、准确的数值算法及其应用

基本信息

批准号：
1720440
负责人：
Jie Shen
金额：
$ 13万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1720440&HistoricalAwards=false
关键词：
Collaborative Research Efficient Stable Accurate

项目摘要

This project focuses on the development of efficient, stable and accurate numerical algorithms for gradient flow systems which are ubiquitous in modeling of real-world phenomena. It is expected that the research will not only lead to efficient numerical algorithms for a class of problems of current interests, but also contribute to better understandings of some fundamental issues in materials science, biotechnology, and other related fields through numerical simulations. This project will also provide opportunities for the involved graduate and undergraduate students to learn critical skills of computational and applied mathematics and to develop state-of-the-art numerical algorithms for science and engineering applications. The free energies of gradient flow systems usually consist of various nonlinear potentials formulated in diverse complex formats which present a major challenge in the construction of efficient and accurate time discretization schemes. The project aims at overcoming this challenge by using a flexible and robust IEQ approach that enables one to develop time discretization schemes for a large class of gradient flow systems. The goals of this proposal are three folds: (i) to develop a unified numerical framework of time-marching schemes for solving general gradient flow models with high nonlinearity; (ii) to develop efficient numerical schemes for a number of challenging gradient flow models of current interests (e.g., nonlinear coupled multivariable models, nonlocal models, anisotropic models, nonlinear coupled systems that follow various physical principles, tensor based liquid crystal models); (iii) to investigate some fundamental issues of viscoelastic drops on substrates and active liquid crystal droplets using the developed predictive numerical tools. The proposed schemes will lead to numerical predictive tools that extend the capability of mathematical and experimental analysis, and contribute to better understanding of some pressing science and engineering issues related to multi-phase complex fluids.

该项目专注于为梯度流系统开发高效、稳定和准确的数值算法，梯度流系统在现实世界现象的建模中普遍存在。预计该研究不仅将为当前感兴趣的一类问题带来有效的数值算法，而且有助于通过数值模拟更好地理解材料科学、生物技术和其他相关领域的一些基本问题。该项目还将为参与的研究生和本科生提供学习计算和应用数学的关键技能以及为科学和工程应用开发最先进的数值算法的机会。梯度流系统的自由能通常由以不同复杂格式表示的各种非线性势组成，这对构建高效且准确的时间离散方案提出了重大挑战。该项目旨在通过使用灵活而强大的 IEQ 方法来克服这一挑战，该方法使人们能够为一大类梯度流系统开发时间离散化方案。该提案的目标有三个：（i）开发时间推进方案的统一数值框架，用于求解具有高非线性的一般梯度流模型； (ii) 为当前感兴趣的许多具有挑战性的梯度流模型（例如，非线性耦合多变量模型、非局部模型、各向异性模型、遵循各种物理原理的非线性耦合系统、基于张量的液晶模型）开发有效的数值方案；（iii）使用开发的预测数值工具研究基底上的粘弹性液滴和活性液晶液滴的一些基本问题。所提出的方案将带来数值预测工具，扩展数学和实验分析的能力，并有助于更好地理解与多相复杂流体相关的一些紧迫的科学和工程问题。