Maximum Bound Principle-Preserving Time Integration Methods for Some Semilinear Parabolic Equations

一些半线性抛物方程的最大有界原理-保时积分方法

• 批准号: 2109633
• 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: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2109633&HistoricalAwards=false
• 关键词: Maximum; Bound; Principle; Preserving; Time

The project will focus on the algorithms and analysis for a class of equations with applications in fluids dynamics, solid mechanics, materials science, chemistry, and cell biology to image and data sciences. Specifically, a class of semilinear parabolic equations will be considered, and the focus will be on ensuring that the computational solutions possess particular important properties called the time-invariant maximum bound principle (MBP). Such properties are important for the models of grain growth and coarsening, thin film microstructure, crystal growth, dislocation-solute interactions, and image restoration and deblurring. The project will design and analyze efficient and accurate MBP-preserving time integration methods of high-order accuracy. The project will develop and disseminate software and provide an interdisciplinary training opportunity for graduate students.The proposed activities contain diverse research topics in computational and applied mathematics, ranging from algorithm design, numerical analysis, and efficient implementation to practical applications in science and engineering. Specifically, the project presents an important step toward developing and analyzing efficient and high-order accurate MBP-preserving time integration methods. Rigorous analysis for a wide class of semilinear parabolic equations within or beyond the current analytical framework will be carried out. This project will not only lead to significant innovations in numerical tools and computer codes for solving these types of equations, but also offer new insights into a number of outstanding theoretical issues on MBP preservation and energy stability in both time-continuous and time-discrete settings. The goals include the design and analysis of linear schemes with high-order accuracy based on the Runge-Kutta integrating factor and the modified scalar auxiliary variable approaches. The project will also extend and develop MBP-preserving time integration methods for some important phase field models beyond the existing analytical framework, including but not limited to, the mass-conserving Allen-Cahn equations with different types of constraints, the convective Allen-Cahn equation and the coupled Navier-Stokes/Allen-Cahn system. These research problems are very useful and challenging with important applications in science and engineering.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.

该项目将重点介绍一类方程的算法和分析，并在流体动力学，固体力学，材料科学，化学和细胞生物学中应用于图像和数据科学。 具体而言，将考虑一类半线性抛物线方程，重点将在确保计算解决方案具有称为时间不变的最大结合原理（MBP）的特定重要属性上。此类特性对于晶粒生长和粗化，薄膜微结构，晶体生长，脱位 - 溶液相互作用以及图像恢复和脱毛很重要。该项目将设计和分析高阶精度的有效，准确的MBP保留时间整合方法。该项目将开发和传播软件，并为研究生提供跨学科的培训机会。拟议的活动包含计算和应用数学的各种研究主题，从算法设计，数值分析以及有效的实施到科学和工程中的实际应用。具体而言，该项目提出了开发和分析有效和高阶准确的MBP保留时间整合方法的重要一步。将对当前分析框架内或超出当前分析框架内或超越一类的半线性抛物线方程进行严格分析。 该项目不仅会导致数值工具和计算机代码的重大创新，以解决这些类型的方程式，而且还提供了有关在时间连续和时间间隔设置中有关MBP保存和能源稳定性的许多出色理论问题的新见解。 目标包括基于runge-kutta集成因子和改进的标量辅助变量方法的线性方案的设计和分析。该项目还将扩展和开发一些重要的相位字段模型的MBP保留时间集成方法，超出了现有的分析框架，包括但不限于具有不同类型约束的质量支持的Allen-CAHN方程，对流的Allen-Cahn方程和couplier-navier-Stokes/Allen-Cahn系统。这些研究问题在科学和工程学中的重要应用非常有用，并且具有挑战性。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。

项目成果

Low regularity integrators for semilinear parabolic equations with maximum bound principles

• DOI: 10.1007/s10543-023-00946-2
• 发表时间: 2022-11
• 期刊: BIT Numerical Mathematics
• 影响因子: 1.5
• 作者: Cao-Kha Doan;Thi-Thao-Phuong Hoang;L. Ju;Katharina Schratz

Trace transfer-based diagonal sweeping domain decomposition method for the Helmholtz equation: Algorithms and convergence analysis

• DOI: 10.1016/j.jcp.2022.110980
• 发表时间: 2020-12
• 期刊: J. Comput. Phys.
• 作者: W. Leng;L. Ju

Operator splitting based structure-preserving numerical schemes for the mass-conserving convective Allen-Cahn equation

• DOI: 10.1016/j.jcp.2022.111695
• 发表时间: 2022-10
• 期刊: J. Comput. Phys.
• 作者: Rihui Lan;Jingwei Li;Yongyong Cai;L. Ju

Unified Solution of Conjugate Fluid and Solid Heat Transfer – Part I. Solid Heat Conduction

共轭流体与固体传热的统一解——第一部分：固体传热

• 发表时间: 2022
• 期刊: Advances in applied mathematics and mechanics
• 影响因子: 1.4
• 作者: Li, Shujie;Ju, Lili
• 通讯作者: Ju, Lili

Parallel exponential time differencing methods for geophysical flow simulations

地球物理流模拟的并行指数时间差分法

• DOI: 10.1016/j.cma.2021.114151
• 发表时间: 2021
• 期刊: Computer Methods in Applied Mechanics and Engineering
• 影响因子: 7.2
• 作者: Lan, Rihui;Leng, Wei;Wang, Zhu;Ju, Lili;Gunzburger, Max
• 通讯作者: Gunzburger, Max