Design and Analysis of Highly Efficient Algorithms for Complex Nonlinear Systems

复杂非线性系统高效算法的设计与分析

基本信息

批准号：
2012585
负责人：
Jie Shen
金额：
$ 29.98万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2022-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012585&HistoricalAwards=false
关键词：
Design Analysis Highly Efficient Algorithms

项目摘要

This project focuses on the development and analysis of innovative, structure preserving algorithms for complex nonlinear systems in science and engineering applications. This project will not only lead to efficient numerical algorithms for a large class of complex nonlinear systems of current interests, but also contribute through numerical simulation to a better understanding of some fundamental issues in materials science, fluid mechanics and other related fields. This project will also provide opportunities for the involved students to learn critical skills of computational and applied mathematics and to develop state-of-the-art numerical algorithms for science and engineering applications. This project will support one graduate student per year.Complex nonlinear systems that possess dissipative or conservation properties are ubiquitous in modeling of real-world phenomena. It is a major challenge to construct efficient and accurate numerical schemes that can preserve important dissipative or conservation properties, and in certain cases, positivity of physical variables. This project will overcome these challenges by extending the flexible and robust scalar auxiliary variable or SAV approach, which has proven to be highly effective for gradient flows. In particular, the SAV approach will be extended to deal with additional difficulties such as those in (i) systems with physical constraints such as mass and/or surface area conservations; (ii) highly anisotropic systems and systems with nonlinear mobilities; (iii) systems with positivity preserving or maximum principles; (iv) systems coupling gradient flows with other conservation laws, and (v) optimizations. The proposed methodology will lead to numerical predictive tools that extend the applicability of mathematical and experimental analysis, and contribute to better understanding of complex physical systems.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.

该项目着重于对科学和工程应用中复杂非线性系统的创新，结构的开发和分析。该项目不仅将为当前利益的一大批复杂的非线性系统提供有效的数值算法，而且还通过数值模拟做出了贡献，从而更好地理解了材料科学，流体力学和其他相关领域的某些基本问题。该项目还将为参与的学生提供机会学习计算和应用数学的关键技能，并为科学和工程应用开发最新的数值算法。该项目将每年支持一名研究生。构建可以保留重要的耗散或保护特性的有效且准确的数值方案以及在某些情况下，是物理变量的阳性，这是一个主要的挑战。该项目将通过扩展灵活且稳健的标量辅助变量或SAV方法来克服这些挑战，该方法已被证明对梯度流非常有效。特别是，将扩展SAV方法，以应对（i）系统中具有物理限制（例如质量和/或表面积保护）的其他困难；（ii）具有非线性迁移率的高度各向异性系统和系统；（iii）具有积极性或最大原则的系统；（iv）系统将梯度流与其他保护法和（v）优化耦合。提出的方法将导致数值预测工具扩展数学和实验分析的适用性，并有助于更好地理解复杂的物理系统。该奖项反映了NSF的法定任务，并认为值得通过基金会的智力优点和更广泛的影响审查标准通过评估来获得支持。