CAREER: High Order Structure-Preserving Numerical Methods for Hyperbolic Conservation Laws

职业：双曲守恒定律的高阶结构保持数值方法

• 批准号: 1753581
• 负责人: Yulong Xing
• 金额: $ 40万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1753581&HistoricalAwards=false
• 关键词: CAREER; Order; Structure; Preserving; Numerical

Partial differential equations of the type known as hyperbolic conservation laws have attracted great attention in mathematical, scientific, and engineering communities due to their wide practical applications in modeling physical systems of interest in fluid mechanics, aerodynamics, meteorology, combustion, and other areas. Development of efficient and accurate numerical algorithms for simulation of solutions to conservation laws continues to be a challenging task. Structure-preserving methods, which provide numerical solutions that preserve a certain continuum property of the underlying models exactly, are recently demonstrated to be more efficient with limited computational resources. This project aims to develop a comprehensive framework to understand structure-preserving methods for hyperbolic conservation laws. The work will have a direct impact in many multi-disciplinary application areas, including fluid and gas dynamics, astrophysics, and atmospheric modeling. This project also has significant broader impact through various educational and outreach activities aimed at students at all levels. These activities include a summer camp program that will expose students including underrepresented minorities to the areas of mathematical modeling, computational science, and computational mathematics. Graduate students will also be mentored and trained through planned working group activities. The notion of conservation (of number, mass, energy, momentum) is a fundamental principle that is used to derive hyperbolic conservation laws. Recent study reveals that structure-preserving numerical methods, which either conserve important physical quantities in addition to mass or preserve other properties of the underlying physical problems, are demonstrated to be more accurate and often have a much improved long time behavior. The objective of this project is to establish a detailed study of novel high-order structure-preserving methods for the linear and nonlinear hyperbolic conservation laws arising in various applications, and to educate students at various levels about the potential and challenges of utilizing numerical simulation to solve important practical problems. The PI aims to study structure-preserving numerical methods in the following directions: (i) Energy conserving methods for wave equations; (ii) Asymptotic preserving methods for kinetic equations; (iii) Well-balanced methods for hyperbolic problems with source terms. The activity is planned to include new algorithm development, theoretical numerical analysis, numerical implementation, and practical applications. This project will also provide excellent training opportunities for graduate and undergraduate students interested in computational sciences, and includes an outreach program for high school students.

被称为双曲守恒定律的偏微分方程在数学、科学和工程界引起了极大的关注，因为它们在流体力学、空气动力学、气象学、燃烧和其他领域的物理系统建模中有广泛的实际应用。开发高效、准确的数值算法来模拟守恒定律的解仍然是一项具有挑战性的任务。结构保留方法提供的数值解能够准确地保留底层模型的某种连续属性，最近被证明在有限的计算资源下更有效。本项目旨在建立一个全面的框架来理解双曲守恒律的结构保存方法。这项工作将对许多多学科应用领域产生直接影响，包括流体和气体动力学、天体物理学和大气建模。该项目还通过针对各级学生的各种教育和推广活动产生了重大的广泛影响。这些活动包括一个夏令营项目，该项目将向学生（包括代表性不足的少数民族）展示数学建模、计算科学和计算数学领域。研究生也将通过计划的工作组活动得到指导和培训。守恒的概念（数量、质量、能量、动量）是用来推导双曲守恒定律的基本原理。最近的研究表明，保留结构的数值方法，既可以保留质量之外的重要物理量，也可以保留潜在物理问题的其他性质，被证明是更准确的，并且通常具有更好的长时间性能。本项目的目的是详细研究各种应用中出现的线性和非线性双曲守恒律的新型高阶结构保持方法，并教育各级学生利用数值模拟解决重要实际问题的潜力和挑战。PI的目标是在以下方向研究保结构数值方法：(i)波动方程的能量守恒方法；（ii）动力学方程的渐近保持方法；带源项的双曲型问题的平衡方法。该活动计划包括新算法开发，理论数值分析，数值实现和实际应用。该项目还将为对计算科学感兴趣的研究生和本科生提供良好的培训机会，并包括一个面向高中生的拓展计划。

项目成果

Well-balanced discontinuous Galerkin methods with hydrostatic reconstruction for the Euler equations with gravitation

具有引力的欧拉方程的静水力重构的良好平衡间断伽辽金法

• DOI: 10.1016/j.jcp.2017.09.063
• 发表时间: 2018
• 期刊: Journal of Computational Physics
• 影响因子: 4.1
• 作者: Gang Li;Yulong Xing
• 通讯作者: Yulong Xing

Convolution Neural Network Shock Detector for Numerical Solution of Conservation Laws

• DOI: 10.4208/cicp.oa-2020-0199
• 发表时间: 2020-06
• 期刊: Communications in Computational Physics
• 影响因子: 3.7
• 作者: Zheng Sun

Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry

球对称中欧拉泊松方程的能量守恒且平衡良好的间断伽辽金方法

• DOI: 10.1093/mnras/stac1257
• 发表时间: 2022
• 期刊: Monthly Notices of the Royal Astronomical Society
• 影响因子: 4.8
• 作者: Zhang, Weijie;Xing, Yulong;Endeve, Eirik
• 通讯作者: Endeve, Eirik

High order well-balanced asymptotic preserving finite difference WENO schemes for the shallow water equations in all Froude numbers

• DOI: 10.1016/j.jcp.2022.111255
• 发表时间: 2021-08
• 期刊: J. Comput. Phys.
• 作者: Guanlan Huang;Y. Xing;T. Xiong

On structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations: Energy conservation and multi-symplecticity

• DOI: 10.1016/j.jcp.2020.109662
• 发表时间: 2019-12
• 期刊: J. Comput. Phys.
• 作者: Zheng Sun;Y. Xing