High Order Schemes: Robustness, Efficiency, and Stochastic Effects

高阶方案：鲁棒性、效率和随机效应

基本信息

批准号：
2010107
负责人：
Chi-Wang Shu
金额：
$ 35万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2010107&HistoricalAwards=false
关键词：
Order Schemes Robustness Efficiency Stochastic

项目摘要

This project concerns algorithm design and analysis of efficient, highly accurate numerical methods for solving partial differential equations. Such equations are used in simulation of systems arising in diverse application fields such as aerospace engineering, semi-conductor device design, astrophysics, and biology. Even with today's fast computers, efficient computational solution of partial differential equations remains a challenge, and it is essential to design improved algorithms that can be used to obtain accurate solutions in these application models. The research aims to produce a suite of powerful computational tools suitable for computer simulations of the complicated solution structure in these applications. The project provides training for a graduate student through involvement in the research.This project conducts research in 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, for solving linear and nonlinear convection-dominated partial differential equations, emphasizing scheme robustness, efficiency, and the treatment of stochastic effects. The project focuses on algorithm development and analysis. Topics of investigation include a new class of multi-resolution WENO schemes with increasingly higher order of accuracy, an inverse Lax-Wendroff procedure for high-order numerical boundary conditions for finite difference schemes on Cartesian meshes solving problems in general geometry, efficient and stable time-stepping techniques for DG schemes and other spatial discretizations, high order accurate bound-preserving schemes and applications, entropy stable DG methods, optimal convergence and superconvergence analysis of DG methods, numerical solutions of stochastic differential equations, and the study of modeling, analysis, and simulation for traffic flow and air pollution. Applications motivate the design of new algorithms or new features in existing algorithms; mathematical tools will be used to analyze these algorithms to give guidelines for their applicability and limitations and to enhance their accuracy, stability, and robustness; and collaborations with engineers and other applied scientists will enable the efficient application of these new 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.

该项目涉及求解偏微分方程的高效、高精度数值方法的算法设计和分析。此类方程用于模拟航空航天工程、半导体器件设计、天体物理学和生物学等不同应用领域中出现的系统。即使使用当今快速的计算机，偏微分方程的有效计算解决方案仍然是一个挑战，并且必须设计可用于在这些应用模型中获得准确解决方案的改进算法。该研究旨在开发一套强大的计算工具，适用于对这些应用中的复杂解决方案结构进行计算机模拟。该项目通过参与研究为研究生提供培训。该项目进行高阶数值方法的算法开发、分析和应用研究，包括不连续伽辽金（DG）有限元方法以及有限差分和有限体积加权本质上非-振荡（WENO）方案，用于求解线性和非线性对流主导的偏微分方程，强调方案的鲁棒性、效率和随机效应的处理。该项目专注于算法开发和分析。研究主题包括一类精度越来越高的新型多分辨率 WENO 方案、笛卡尔网格上有限差分方案的高阶数值边界条件的逆 Lax-Wendroff 过程，解决一般几何中的问题、高效且稳定的时间-DG方案和其他空间离散化的步进技术、高阶精确保界方案和应用、熵稳定DG方法、DG方法的最优收敛和超收敛分析、DG方法的数值解随机微分方程，以及交通流和空气污染的建模、分析和模拟研究。应用激发了新算法的设计或现有算法的新功能；将使用数学工具来分析这些算法，为其适用性和局限性提供指导，并提高其准确性、稳定性和鲁棒性；与工程师和其他应用科学家的合作将使这些新算法得到有效应用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。