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

• 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

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.

该项目的重点是在科学和工程应用中复杂非线性系统的创新，结构保持算法的开发和分析。该项目不仅将为当前感兴趣的一大类复杂非线性系统提供有效的数值算法，而且还将通过数值模拟更好地理解材料科学，流体力学和其他相关领域的一些基本问题。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先进的数值算法。本计画每年将资助一位研究生。复杂的非线性系统具有耗散或守恒性质，在现实世界现象的模拟中是普遍存在的。这是一个重大的挑战，以建立有效和准确的数值方案，可以保持重要的耗散或守恒性质，并在某些情况下，积极的物理变量。该项目将通过扩展灵活和强大的标量辅助变量或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，SAV方法将扩展到处理额外的困难，如在（i）系统的物理约束，如质量和/或表面积守恒;（ii）高度各向异性系统和系统的非线性迁移率;（iii）系统的正性保持或最大值原则;（iv）系统耦合梯度流与其他守恒定律，和（v）优化。建议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，并有助于更好地了解复杂的物理system.This奖项反映了NSF的法定使命，并已被认为是值得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的支持。

项目成果

Efficient and Unconditional Energy Stable Schemes for the Micropolar Navier-Stokes Equations

• DOI: 10.4208/csiam-am.so-2021-0008
• 发表时间: 2022-06
• 期刊: CSIAM Transactions on Applied Mathematics
• 作者: Jie Shen null;Nan Zheng

Bound/positivity preserving and unconditionally stable schemes for a class of fourth order nonlinear equations

一类四阶非线性方程的有界/正性保持和无条件稳定格式

• DOI: 10.1016/j.jcp.2022.111177
• 发表时间: 2022
• 期刊: Journal of Computational Physics
• 影响因子: 4.1
• 作者: Huang, Fukeng;Shen, Jie;Wu, Ke
• 通讯作者: Wu, Ke

Generalized SAV approaches for gradient systems

• DOI: 10.1016/j.cam.2021.113532
• 发表时间: 2021-03
• 期刊: J. Comput. Appl. Math.
• 作者: Q. Cheng;Chun Liu;Jie Shen

A generalized SAV approach with relaxation for dissipative systems

• DOI: 10.1016/j.jcp.2022.111311
• 发表时间: 2022-01
• 期刊: J. Comput. Phys.
• 作者: Yanrong Zhang;Jie Shen

Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation

广义AllenCahn方程高阶有限差分格式的离散极大值原理

• DOI: 10.4310/cms.2022.v20.n5.a9
• 发表时间: 2022
• 期刊: Communications in Mathematical Sciences
• 影响因子: 1
• 作者: Shen, Jie;Zhang, Xiangxiong
• 通讯作者: Zhang, Xiangxiong