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

该项目的目的是为浅水方程和其他带源项的双曲型守恒律提供一类新的高精度、高效率、平衡良好的间断Galerkin(DG)格式和加权基本无振荡(WENO)格式。拟议的活动包括新算法开发、理论数值分析、数值实施问题和实际应用的全面覆盖。研究人员建议从以下几个方面对高效的高阶良好平衡方法进行详细的研究：1.良好平衡方法的发展：将对不同地区出现的几个方程发展非常精确的良好平衡数值方法；2.浅水方程：将发展针对浅水方程的正性保持良好平衡方法。然后，研究人员将考察它们的性能，包括效率、可扩展性等，并研究它们在沿海海洋模拟中的潜在应用；3.引力场下的欧拉方程：引力场中的流体动力学演化存在于大多数天体物理问题中。研究人员将发展这类系统的平衡方法；4.非线性水波方程：将发展守恒的DG方法来求解非线性色散波动方程。该项目将为求解浅水方程和其他带源项的守恒律提供更有效和准确的数值方法。它将对水利工程和大气模拟中的许多应用问题产生直接影响，并适用于化学、生物学、流体力学、天体物理和气象学中的其他源项问题。由于其多学科性质，拟议的研究将启动并作为与应用数学家、水利工程师和天体物理学家合作研究工作的坚实基础，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。

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.