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.

该项目将为具有不确定性的双曲守恒定律/平衡律的新型随机模型和数值算法的设计和分析做出重大贡献。此类系统是模拟各种复杂物理现象（包括波传播和流体流动）的基本数学工具。开发的随机模型和数值方法将提高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，其应用范围从沿海和水利工程到模拟大气和海洋现象，包括飓风、台风、海啸和由此产生的风暴潮。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 将参与外展活动，并将继续致力于增加 STEM 的多样性和扩大参与。该项目的主要目标是开发和分析鲁棒的高分辨率结构保持随机模型和不确定性双曲守恒/平衡定律的数值方法。作为主要范例，研究将重点关注浅水方程，但设计的工具将适用于更广泛的守恒定律/平衡定律，以及对流扩散模型问题，以及比浅水更普遍的问题将研究方程。浅水模型和相关系统广泛应用于与地表流动力学建模和预测相关的许多重要应用，例如河流、湖泊和沿海地区的水流。确定性浅水方程的经典系统，称为圣维南系统，是守恒定律/平衡律的非线性双曲系统。圣维南模型可以接受非光滑解，这些解可能具有冲击、稀疏波，并且如果底部地形不连续，则可能存在接触不连续性。在后一种情况下，解决方案可能不是唯一的，这使得即使在一维确定性情况下开发准确有效的算法也更具挑战性。一方面，考虑科里奥利力、底部摩擦应力和数据中的随机性/不确定性等因素的影响对于设计具有改进的预测能力的模型和模拟至关重要。另一方面，此类数学模型可能对构建稳健的数值算法提出重大挑战。因此，本研究的主要目标是（1）开发侵入式和非侵入式鲁棒不确定性量化（UQ）技术，从而产生物理相关的随机浅水模型和相关系统； (2) 为结果模型设计和分析自适应高阶精确结构保持确定性和随机求解器； (3) 并开发计算高效且可并行的算法。该项目取得的进展将解决非线性守恒定律/平衡律数值方法和昆士兰大学交通问题数值方法方面的突出挑战。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。