Development of high-order accurate numerical methods for the shallow-water equations and other hyperbolic conversation laws with source terms

开发浅水方程和其他带有源项的双曲对话律的高阶精确数值方法

• 批准号: 1216454
• 负责人: Yulong Xing
• 金额: $ 16.62万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1216454&HistoricalAwards=false
• 关键词: Development; order; accurate; numerical; methods

The objective of the proposed project is to provide a class of novel high-order accurate and efficient well-balanced discontinuous Galerkin (DG) and Weighted Essentially Non-Oscillatory (WENO) schemes for the shallow-water equations and other hyperbolic conservation laws with source terms. The proposed activity includes a comprehensive coverage of new algorithm development, theoretical numerical analysis, numerical implementation issues and practical applications. The investigator proposes to provide a detailed study of highly efficient high-order well-balanced methods in the following directions: 1. Development of well-balanced methods: Very accurate well-balanced numerical methods will be developed for several equations arising in different areas; 2. Shallow-water equations: Positivity-preserving well-balanced methods for the shallow-water equations will be developed. Then, the investigator will investigate their performance, including efficiency, scalability, etc., and study their potential application in the coastal ocean modeling; 3. Euler equations under a gravitational field: Hydrodynamical evolution in a gravitational field arises in most astrophysical problems. The investigator will develop well-balanced methods for such system; 4. Nonlinear water wave equations: Conservative DG methods will be developed for nonlinear dispersive wave equations.The proposed project will provide more efficient and accurate numerical approaches to solve the shallow-water equations, and other conservation laws with source term. It will have a direct impact in many application problems arising from hydraulic engineering and atmospheric modeling, and is suitable for other source-term problems in chemistry, biology, fluid dynamics, astrophysics, and meteorology. Due to its multi-disciplinary nature, the proposed research will initiate and serve as a solid foundation for collaborative research work with applied mathematicians, hydraulic engineers and astrophysicists, and promote interdisciplinary research between Oak Ridge National Laboratory and the University of Tennessee. The proposed project will also provide training and education opportunities for both graduate and undergraduate students interested in computational mathematics.

该项目的目标是为浅水方程和其他双曲守恒定律提供一类新颖的高阶精确且高效的良好平衡的不连续伽辽金（DG）和加权本质非振荡（WENO）格式条款。拟议的活动全面涵盖新算法开发、理论数值分析、数值实现问题和实际应用。研究者建议在以下方向对高效高阶平衡方法进行详细研究： 1. 平衡方法的发展：针对不同领域中出现的多个方程开发非常精确的平衡数值方法； 2. 浅水方程：将开发浅水方程的保正性平衡方法。然后，研究者将调查它们的性能，包括效率、可扩展性等，并研究它们在沿海海洋建模中的潜在应用； 3. 引力场下的欧拉方程：引力场中的流体动力学演化出现在大多数天体物理问题中。研究者将为该系统开发均衡的方法； 4. 非线性水波方程：将针对非线性色散波动方程开发保守的DG方法。该项目将为求解浅水方程和其他带有源项的守恒定律提供更高效、更精确的数值方法。它将对水利工程和大气建模产生的许多应用问题产生直接影响，并且适用于化学、生物学、流体动力学、天体物理学和气象学等其他源项问题。由于其多学科性质，拟议的研究将启动并为与应用数学家、水利工程师和天体物理学家的合作研究工作奠定坚实的基础，并促进橡树岭国家实验室和田纳西大学之间的跨学科研究。拟议的项目还将为对计算数学感兴趣的研究生和本科生提供培训和教育机会。