Structure-Preserving Algorithms for Hyperbolic Balance Laws with Uncertainty

该项目将为设计和分析具有不确定性的双曲守恒/平衡定律的新型随机模型和数值算法做出重大贡献。这些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不同科学和工程领域使用的计算工具的准确性和预测能力，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。为了培养下一代数学人才，除了指导研究生和本科生外，pi还将参与外展活动，并将继续努力增加STEM的多样性和扩大参与范围。该项目的主要目标是开发和分析具有不确定性的双曲守恒/平衡定律的鲁棒高分辨率结构保持随机模型和数值方法。作为一个主要范例，研究将集中在浅水方程上，但设计的工具将适用于更广泛的守恒/平衡定律，以及对流-扩散模型问题，并且将研究比浅水方程更一般的问题。浅水模型及其相关系统广泛应用于许多与地表流动动力学建模和预测相关的重要应用中，例如河流、湖泊和沿海地区的水流。经典的确定性浅水方程组，被称为圣维南方程组，是一个守恒/平衡定律的非线性双曲系统。Saint-Venant模型可以接受非光滑解，这些解可能有冲击、稀疏波，如果底部地形不连续，则存在接触不连续。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，考虑到科里奥利力、底部摩擦应力和数据中的随机性/不确定性的影响，对于设计具有改进预测能力的模型和模拟至关重要。另一方面，这样的数学模型对鲁棒数值算法的构建提出了重大挑战。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，从而建立与物理相关的随机浅水模型和相关系统；(2)设计和分析结果模型的自适应高阶精确保结构确定性和随机解算器；(3)开发计算效率高、可并行化的算法。该项目取得的进展将解决非线性守恒/平衡定律的数值方法和运输问题的UQ方面的突出挑战。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.