Development and Application of Modern Numerical Methods for Nonlinear Hyperbolic Systems of Partial Differential Equations

偏微分方程非线性双曲型系统现代数值方法的发展与应用

基本信息

批准号：
2208438
负责人：
Alina Chertock
金额：
$ 36.86万
依托单位：
North Carolina State University
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：
2022-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208438&HistoricalAwards=false
关键词：
Development Application Modern Numerical Methods

项目摘要

The goal of this project is to develop new mathematical and computational tools for a large class of hyperbolic conservation and balance laws and related problems. Such systems arise in a wide variety of applications ranging from classical fluid dynamics (gas dynamics including multicomponent and multiphase compressible flows, many shallow water models including rotating shallow water equations, thermal rotating shallow water equations, shallow magnetohydrodynamic equations, and others), astrophysics, meteorology, oceanography, atmospheric sciences, to electromagnetism and modern biological models. While the PI is planning to work on several particular applications, the main focus of the research will be in the development of novel numerical methods and computational techniques that can be applied to a wide class of applied problems arising in today’s science. The project has also a potential to contribute to the emergence of accurate, robust and efficient algorithms and will overall increase the practical applicability on numerical methods. This project involves the training of graduate students. Many practical applications, especially in the cases of high space dimensions, require development and implementation of special numerical methods that are not only consistent with the governing system of partial differential equations, but also preserve certain structural and asymptotic properties of the underlying problem at the discrete level. The project is aimed at the development of efficient high-order methods for systems of conservation and balance laws whose basic properties go beyond consistency, stability and convergence. This will be achieved by designing special numerical techniques for (i) finding a delicate balance between the numerical diffusion and dispersion to ensure sharp—yet non-oscillatory—resolution of shock and contact waves, while achieving a high order accuracy in smooth regions, (ii) exactly preserving physically relevant steady-state solutions and involution constraints, (iii) establishing asymptotic preserving properties in certain stiff regimes, and (iv) analyzing the influence of uncertainties in problems with random data. The design and implementation of the new numerical schemes will be based on high-order shock-capturing finite-volume and finite-difference methods, accurate and efficient time integrators, and stochastic Galerkin and collocation methods, utilizing the main advantages of each of these methods in the context of the studied problems. The derivation of such numerical techniques is fundamental for understanding many physical phenomena and will contribute to their quantitative and qualitative study.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.

该项目的目标是为一大类双曲守恒定律和平衡定律及相关问题开发新的数学和计算工具，此类系统出现在从经典流体动力学（气体动力学，包括多组分和多相可压缩流）的各种应用中。），许多浅水模型，包括旋转浅水方程，热旋转浅水方程，浅磁流体动力学方程等），天体物理学，气象学，海洋学，大气科学，电磁学和现代生物模型。 PI 计划致力于几个特定的应用，研究的主要重点将是开发新颖的数值方法和计算技术，这些方法和计算技术可应用于当今科学中出现的各种应用问题。该项目也具有潜力。有助于出现准确、稳健和高效的算法，并将全面提高数值方法的实际适用性。该项目涉及研究生的培训，特别是在高空间维度的情况下，需要开发和实施。特殊的数值方法不仅符合治理体系该项目旨在为守恒定律和平衡律系统开发有效的高阶方法，其基本属性超出了一致性、稳定性和稳定性。这将通过设计特殊的数值技术来实现：(i) 找到数值扩散和色散之间的微妙平衡，以确保冲击波和接触波的锐利但非振荡分辨率，同时在平滑区域实现高阶精度。，（二）精确地保持物理相关的稳态解和对合约束，（iii）在某些刚性状态下建立渐近保持性质，以及（iv）分析随机数据问题中的不确定性的影响。基于高阶冲击捕获有限体积和有限差分方法、精确高效的时间积分器以及随机伽辽金和搭配方法，在研究的背景下利用这些方法的主要优点这种数值技术的推导对于理解许多物理现象至关重要，并将有助于其定量和定性研究。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。。