CAREER: Tailored Entropy Stable Discretizations of Nonlinear Conservation Laws

流体流动的模拟是许多科学领域的基础，从环境和航空航天工程到太阳物理学。然而，使用现有工具进行下一代建模和分析在计算上具有挑战性。量身定制的数值方法有可能解决这些限制。例如，基于投影的降阶模型通过用更便宜的低维代理替换高保真度模型，减少了与许多查询场景（如工程设计或不确定性量化）相关的计算成本。同样，高阶精确格式在解决瞬态涡流的精细尺度特征方面特别有效。不幸的是，当应用于流体动力学方程时，这些数值方法会经历非物理不稳定性，这可能导致模拟意外失败。这个项目的目标是使鲁棒和有效的模拟使用离散的“熵稳定”方案。通过将基本能量原理直接构建到离散化中，熵稳定方法在继承可验证属性的同时保持准确性，从而提高“开箱即用”的鲁棒性。非线性守恒律的离散熵稳定高阶离散化在过去7年中得到了迅速的发展。本项目将把该方法扩展到现有方法不理想或不可用的三个领域：(1)非一致性网格的高阶方法，(2)具有精细尺度特征的域边界的高阶物理帧离散化，以及(3)降阶建模。此外，PI将把拟议研究的各个方面与一个旨在促进计算科学和提高大学生和K-12教师保留率的教育计划结合起来。具体来说，PI将(1)为工程本科生设计和监督高级顶点研究项目，(2)为K-12教师组织一个以数值建模和基于发现的学习为中心的暑期研究项目。夏季研究项目还将为研究生提供指导经验，PI将与教师合作，将数值计算的概念纳入可重复使用的课堂模块。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 simulation of fluid flow is foundational to many scientific fields, ranging from environmental and aerospace engineering to solar physics. However, next-generation modeling and analysis is computationally challenging using existing tools. Tailored numerical methods have the potential to address such limitations. For example, projection-based reduced order models decrease computational costs associated with many-query scenarios (such as engineering design or uncertainty quantification) by replacing a high fidelity model with a less expensive low-dimensional surrogate. Similarly, high order accurate schemes are particularly effective at resolving fine-scale features in transient vorticular flows. Unfortunately, when applied to the equations of fluid dynamics, these numerical methods experience non-physical instabilities which can cause simulations to fail unexpectedly. The goal of this project is to enable robust and efficient simulations using discretely "entropy stable" schemes. By building fundamental energetic principles directly into a discretization, entropy stable methods retain accuracy while inheriting verifiable properties which improve "out-of-the-box" robustness. Discretely entropy stable high order discretizations for nonlinear conservation laws have seen rapid development over the last 7 years. This project will extend this methodology to three areas where existing approaches are suboptimal or unavailable: (1) high order methods on non-conforming meshes, (2) high order physical-frame discretizations for domain boundaries with fine-scale features, and (3) reduced order modeling. Additionally, the PI will integrate aspects of the proposed research with an educational program aimed at promoting computational science and improving retention among college students and K-12 teachers. Specifically, the PI will (1) design and supervise senior capstone research projects for engineering undergraduates, and (2) organize a summer research program for K-12 teachers centered around numerical modeling and discovery-based learning. The summer research program will also provide graduate students with mentoring experience, and the PI will follow up by partnering with teachers to incorporate concepts from numerical computing into reusable classroom modules.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.