| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | April 26, 2016 |
| Latest Amendment Date: | April 26, 2016 |
| Award Number: | 1621111 |
| Award Instrument: | Standard Grant |
| Program Manager: |
rosemary renaut
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 1, 2015 |
| End Date: | July 31, 2016 (Estimated) |
| Total Intended Award Amount: | $63,981.00 |
| Total Awarded Amount to Date: | $63,981.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
200 UNIVERSTY OFC BUILDING RIVERSIDE CA US 92521-0001 (951)827-5535 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
CA US 92521-0001 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | COMPUTATIONAL MATHEMATICS |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
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.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
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 (SWEs) and other hyperbolic conservation laws with source terms. During the funded period, we have successfully developed new well-balanced positivity-preserving DG methods for the two-dimensional SWEs to address two challenges in their hydrology application: non-flat bottom and wetting and drying front. The proposed DG methods on unstructured triangular meshes have been implemented and tested on parallel machines and nice numerical performance has been observed. We also proposed the first well-balanced finite element method for the SWEs which can preserve the general moving water equilibrium steady state, and capture the small perturbation to such system well. For the SWEs through channels with irregular geometry, we design high order finite volume WENO methods which also have the positivity-preserving feature. These results will lead to efficient numerical simulation of the SWEs in their hydraulic applications, and we have studied the extension of our methods for tidal bores problem with engineers. The other mathematical model under consideration is the Euler equation in a gravitational field. We have designed both DG and WENO methods that can preserve not only the isothermal but also the polytropic hydrostatic balance exactly. These methods have great potential in the related astrophysical applications. In addition, we designed energy conserving DG methods for various linear and nonlinear wave equations including the second order wave equation, generalized Korteweg-de Vries equation, nonlinear Camassa-Holm equation and nonlinear Schrodinger equations. Detailed numerical analysis has been carried out to verify their convergence rate.
Our methods have great potential to be utilized in hydraulic, climate and astrophysical applications. 19 papers have been published or accepted for publication in peer-reviewed journals as a result of this project. Other broader impact of this project includes the training and education of graduate and undergraduate students including the woman and underrepresented minority students. The project has been presented at many conferences and workshops. I have taught both graduate and undergraduate courses in numerical analysis, and some results from this project have been used in the class to promote the students’ interest in computational mathematics.
Last Modified: 10/23/2016
Modified by: Yulong Xing
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