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

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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.