CAREER: High Order Structure-Preserving Numerical Methods for Hyperbolic Conservation Laws

被称为双曲型守恒律的偏微分方程因其在流体力学、空气动力学、气象学、燃烧和其他领域的物理系统建模中的广泛应用而引起了数学、科学和工程界的极大关注。开发高效而准确的数值算法来模拟守恒律的解仍然是一项具有挑战性的任务。结构保持法是一种用有限的计算资源提供数值解的方法，它精确地保持了基本模型的某些连续统性质，是一种更有效的方法。这个项目旨在开发一个全面的框架来理解双曲型守恒律的结构保持方法。这项工作将对许多多学科应用领域产生直接影响，包括流体和气体动力学、天体物理学和大气建模。该项目还通过针对各级学生的各种教育和外联活动产生了重大而广泛的影响。这些活动包括一个夏令营计划，该计划将使学生，包括代表人数较少的少数族裔，接触数学建模、计算科学和计算数学领域。还将通过计划的工作组活动对研究生进行指导和培训。守恒的概念(数量、质量、能量、动量)是用来推导双曲守恒定律的基本原理。最近的研究表明，保持结构的数值方法，既保留了重要的物理量又保留了质量，或者保留了潜在物理问题的其他性质，被证明是更准确的，并且通常具有更好的长期行为。这个项目的目的是建立一个新的高阶结构保持方法的详细研究，线性和非线性双曲型守恒律在各种应用中出现，并教育不同层次的学生利用数值模拟来解决重要的实际问题的潜力和挑战。PI主要研究如下方向的保结构数值方法：(I)波动方程的能量守恒方法；(Ii)动力学方程的渐近保持方法；(Iii)带源项的双曲型问题的良好平衡方法。该活动计划包括新算法开发、理论数值分析、数值实施和实际应用。该项目还将为对计算科学感兴趣的研究生和本科生提供极好的培训机会，并包括一个面向高中生的外展计划。

Partial differential equations of the type known as hyperbolic conservation laws have attracted great attention in mathematical, scientific, and engineering communities due to their wide practical applications in modeling physical systems of interest in fluid mechanics, aerodynamics, meteorology, combustion, and other areas. Development of efficient and accurate numerical algorithms for simulation of solutions to conservation laws continues to be a challenging task. Structure-preserving methods, which provide numerical solutions that preserve a certain continuum property of the underlying models exactly, are recently demonstrated to be more efficient with limited computational resources. This project aims to develop a comprehensive framework to understand structure-preserving methods for hyperbolic conservation laws. The work will have a direct impact in many multi-disciplinary application areas, including fluid and gas dynamics, astrophysics, and atmospheric modeling. This project also has significant broader impact through various educational and outreach activities aimed at students at all levels. These activities include a summer camp program that will expose students including underrepresented minorities to the areas of mathematical modeling, computational science, and computational mathematics. Graduate students will also be mentored and trained through planned working group activities. The notion of conservation (of number, mass, energy, momentum) is a fundamental principle that is used to derive hyperbolic conservation laws. Recent study reveals that structure-preserving numerical methods, which either conserve important physical quantities in addition to mass or preserve other properties of the underlying physical problems, are demonstrated to be more accurate and often have a much improved long time behavior. The objective of this project is to establish a detailed study of novel high-order structure-preserving methods for the linear and nonlinear hyperbolic conservation laws arising in various applications, and to educate students at various levels about the potential and challenges of utilizing numerical simulation to solve important practical problems. The PI aims to study structure-preserving numerical methods in the following directions: (i) Energy conserving methods for wave equations; (ii) Asymptotic preserving methods for kinetic equations; (iii) Well-balanced methods for hyperbolic problems with source terms. The activity is planned to include new algorithm development, theoretical numerical analysis, numerical implementation, and practical applications. This project will also provide excellent training opportunities for graduate and undergraduate students interested in computational sciences, and includes an outreach program for high school students.