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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.

这个项目的目标是为一大类双曲守恒和平衡定律以及相关问题开发新的数学和计算工具。这种系统出现在从经典流体动力学（包括多组分和多相可压缩流的气体动力学，包括旋转浅水方程、热旋转浅水方程、浅水磁流体动力学方程等的许多浅水模型）、天体物理学、气象学、海洋学、大气科学到电磁学和现代生物模型的各种应用中。虽然PI计划在几个特定的应用程序上工作，但研究的主要重点将是开发新的数值方法和计算技术，这些方法和技术可以应用于当今科学中出现的广泛的应用问题。该项目也有可能有助于准确，强大和高效的算法的出现，并将全面提高数值方法的实用性。这个项目涉及研究生的培训。许多实际应用，特别是在高空间维数的情况下，需要开发和实施特殊的数值方法，不仅符合偏微分方程的控制系统，但也保持一定的结构和渐近性质的基本问题在离散水平。该项目的目的是为基本性质超越一致性、稳定性和收敛性的守恒律和平衡律系统开发有效的高阶方法。这将通过设计特殊的数值技术来实现：（i）在数值扩散和色散之间找到微妙的平衡，以确保冲击波和接触波的尖锐而非振荡的分辨率，同时在光滑区域实现高阶精度，（ii）精确地保持物理相关的稳态解和对合约束，（iii）在某些刚性区域建立渐近保持特性，以及（iv）分析随机数据问题中不确定性的影响。新的数值格式的设计和实施将基于高阶激波捕获有限体积和有限差分法、精确有效的时间积分器以及随机Galerkin和配置方法，利用这些方法在所研究问题的背景下的主要优点。这种数值技术的推导是理解许多物理现象的基础，并将有助于它们的定量和定性研究。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。