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

• 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; Development; Application; Modern; Numerical; Methods

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.

该项目的目的是为大量的双曲线保护以及平衡法律和相关问题开发新的数学和计算工具。此类系统出现在各种应用中，包括经典的流体动力学（包括多组分和多相兼容流，包括旋转浅水方程，热旋转浅水方程，浅磁性水力动力学方程以及其他许多浅水模型），天文学，气象学，海洋学，大气层，电子和现代化。尽管PI计划在几种特定的应用程序上工作，但研究的主要重点将是开发新型的数值方法和计算技术，这些方法可以应用于当今科学中引起的广泛应用问题。该项目还具有有助于准确性，鲁棒和有效算法的出现的潜力，并将总体上增加对数值方法的实际应用。该项目涉及研究生的培训。许多实际应用，尤其是在较高空间维度的情况下，都需要开发和实施特殊的数值方法，这些方法不仅与偏微分方程的管理系统一致，而且还保留在离散级别上基本问题的某些结构和不对称属性。该项目旨在开发有效的高阶方法，用于保护和平衡法律的基本属性超出一致性，稳定性和收敛性。 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 asymmetric preserving properties in certain stiff regimes, and （iv）分析随机数据问题中不确定性的影响。新的数值方案的设计和实现将基于高级冲击捕获有限量和有限差异方法，准确有效的时间积分器，以及随机的Galerkin和Politocation方法，使用这些方法的主要优势，在研究问题问题的上下文中。这种数值技术的推导对于理解许多物理现象至关重要，并将为其定量和定性研究做出贡献。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的审查标准来通过评估来获得的支持。