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High Order Schemes: Bound Preserving, Moving Boundary, Stochastic Effects and Efficient Time Discretization

高阶方案：保界、移动边界、随机效应和高效时间离散化

基本信息

批准号：
2309249
负责人：
Chi-Wang Shu
金额：
$ 40万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-07-01 至 2026-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309249&HistoricalAwards=false
关键词：
Order Schemes Bound Preserving Moving

项目摘要

The project aims to develop efficient and high-precision numerical methods for solving partial differential equations in various important scientific and engineering applications, such as aerospace engineering, semiconductor device design, astrophysics, and biological applications. Even with today's fast supercomputers, it is still essential to design efficient and reliable algorithms to obtain accurate solutions to these applications where high precision can improve the safety and performance of those devices. These algorithms will make positive contributions to computer simulations of the complicated solution structure in these applications. The project will include workforce development for students from underrepresented groups in STEM.The project aims to investigate algorithm development, analysis, and application of high-order numerical methods, including discontinuous Galerkin (DG) finite element methods and finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes. The algorithms will be designed to solve linear and nonlinear convection-dominated partial differential equations (PDEs), emphasizing bound preserving, moving boundary, stochastic effects and efficient time discretization. Topics of the research investigations will include an inverse Lax-Wendroff procedure for numerical boundary conditions with moving boundaries and interfaces, mathematical properties and efficient solvers for forward-backward coupled PDE systems from traffic flow modeling, high order numerical methods for hysteretic flows, robust high order Lagrangian methods, efficient and stable time-stepping techniques for DG schemes and other spatial discretizations, high order accurate bound-preserving schemes and applications including problems involving highly nonlinear constraints and one step Lax-Wendroff type time discretizations, problems with stiff source terms, high order DG schemes for stationary hyperbolic equations and radiative transfer equations, oscillation-free DG methods, and numerical solutions of stochastic differential equations. The research will provide guidelines for the algorithms' applicability and limitations while enhancing their accuracy, stability, and robustness. The research will include collaborations with engineers and other applied scientists to enable the efficient application of these new algorithms or new features in existing algorithms.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.

该项目旨在开发高效和高精度的数值方法，用于解决各种重要科学和工程应用中的偏微分方程，如航空航天工程，半导体器件设计，天体物理学和生物学应用。即使有了当今快速的超级计算机，设计高效可靠的算法仍然至关重要，以获得这些应用的精确解决方案，其中高精度可以提高这些设备的安全性和性能。这些算法将为这些应用中复杂解结构的计算机模拟做出积极的贡献。该项目将包括STEM中代表性不足的学生的劳动力发展。该项目旨在研究高阶数值方法的算法开发，分析和应用，包括间断Galerkin（DG）有限元方法和有限差分和有限体积加权基本无振荡（韦诺）格式。这些算法将被设计用于求解线性和非线性对流占优偏微分方程（PDE），强调边界保持，移动边界，随机效应和有效的时间离散化。研究调查的主题将包括移动边界和界面的数值边界条件的逆Lax-Wendroff程序，交通流建模的前后耦合PDE系统的数学特性和有效求解器，滞后流的高阶数值方法，鲁棒高阶拉格朗日方法，DG计划和其他空间离散的有效和稳定的时间步长技术，高阶精度保界格式及其应用，包括高度非线性约束和一步Lax-Wendroff型时间离散问题，刚性源项问题，定常双曲方程和辐射传输方程的高阶DG格式，无振荡DG方法，以及随机微分方程的数值解。该研究将为算法的适用性和局限性提供指导，同时提高其准确性，稳定性和鲁棒性。该研究将包括与工程师和其他应用科学家的合作，以使这些新算法或现有算法中的新功能得到有效应用。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。