Structure-Preserving Algorithms for Hyperbolic Balance Laws with Uncertainty

不确定性双曲平衡定律的结构保持算法

基本信息

批准号：
2207207
负责人：
Yekaterina Epshteyn
金额：
$ 45万
依托单位：
University of Utah
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2207207&HistoricalAwards=false
关键词：
Structure Preserving Algorithms Hyperbolic Balance

项目摘要

The project will make significant contributions to the design and analysis of novel stochastic models and numerical algorithms for hyperbolic conservation/balance laws with uncertainty. Such systems are the essential mathematical apparatus for modeling a variety of complex physical phenomena, including wave propagation and fluid flow. The developed stochastic models and numerical methods will improve the accuracy and predictive capabilities of the computational tools used in different areas of science and engineering with applications ranging from coastal and hydraulic engineering, to modeling atmospheric and oceanographic phenomena, including hurricanes, typhoons, tsunamis, and resulting storm surges. The obtained numerical algorithms and data will be made available to other researchers. For the training of the next-generation mathematical workforce, in addition to mentoring of graduate and undergraduate students, the PIs will participate in outreach activities and will continue to work towards increasing diversity and broadening participation within STEM.The main objective of the project is the development and analysis of robust high-resolution structure-preserving stochastic models and numerical methods for hyperbolic conservation/balance laws with uncertainty. As a primary exemplar, the research will focus on the shallow water equations, but the designed tools will be applicable to a wider class of conservation/balance laws, as well as to convection-diffusion model problems, and problems more general than the shallow water equations will be investigated. Shallow water models and related systems are widely used in many important applications related to modeling and prediction of the dynamics of surface flows, such as water flows in rivers, lakes, and coastal areas. The classical system of deterministic shallow water equations, known as the Saint-Venant system, is a nonlinear hyperbolic system of conservation/balance laws. The Saint-Venant model can admit non-smooth solutions that may have shocks, rarefaction waves, and if the bottom topography is discontinuous, contact discontinuities. In the latter case, the solution may not be unique, which makes the development of accurate and efficient algorithms more challenging even in the one-dimensional deterministic case. Taking into account the effects of, for example, Coriolis forces, bottom friction stresses, and randomness/uncertainties in the data, on one hand is crucial for the design of models and simulations with improved predictive capabilities. On the other hand, such mathematical models can present a significant challenge for the construction of robust numerical algorithms. Therefore, the primary goals of this research are (1) to develop intrusive and non-intrusive robust uncertainty quantification (UQ) techniques that will lead to physically-relevant stochastic shallow water models and related systems; (2) to design and analyze adaptive high-order accurate structure-preserving deterministic and stochastic solvers for resulting models; (3) and to develop computationally efficient and parallelizable algorithms. Advances achieved by the project will tackle outstanding challenges in numerical methods for nonlinear conservation/balance laws and UQ for transport problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将对新颖的随机模型的设计和分析和分析，并分析具有不确定性双曲线保护/平衡定律的数值算法。这样的系统是用于建模各种复杂物理现象的必不可少的数学设备，包括波传播和流体流动。开发的随机模型和数值方法将提高在科学和工程不同领域使用的计算工具的准确性和预测能力，其应用包括沿海和液压工程，再到建模大气和海洋学现象，包括飓风，Typhoons，Typhoons，Tsunamis以及导致风暴Surges。获得的数值算法和数据将提供给其他研究人员。为了培训下一代数学劳动力，除了指导研究生和本科生外，PI还将参加外展活动，并将继续致力于增加多样性并扩大STEM内的参与。作为主要的典范，该研究将集中在浅水方程上，但是设计的工具将适用于更广泛的保护/平衡法，以及对流扩散模型问题，并且比浅水方程更一般的问题将被调查。浅水模型和相关系统广泛用于与表面流动动力学的建模和预测有关的许多重要应用，例如河流，湖泊和沿海地区的水流。确定性浅水方程式的经典系统，即圣人系统，是一种非线性的保护/平衡法系统。 Saint-Venant模型可以接收可能具有冲击波，稀有波的非平滑溶液，如果底部地形是不连续的，则接触不连续性。在后一种情况下，解决方案可能不是唯一的，这使得准确有效的算法的发展即使在一维确定性情况下也更具挑战性。考虑到数据中的科里奥利力，底部摩擦应力以及数据中的随机性/不确定性的影响，一方面对于具有提高预测能力的模型和模拟的设计至关重要。另一方面，这种数学模型对稳健数值算法的构建构成了重大挑战。因此，这项研究的主要目标是（1）开发侵入性和非侵入性的鲁棒不确定性定量（UQ）技术，该技术将导致物理上与身体相关的随机浅水模型和相关系统；（2）设计和分析生成模型的自适应高阶精确结构确定性和随机求解器；（3）并开发计算上有效且可行的算法。该项目所取得的进步将解决非线性保护/平衡法和运输问题UQ的数值方法中的杰出挑战。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估来审查标准的评估。