Study of Limiting Methods for Computation of Conservation Laws and Other Hyperbolic Problems

许多自然系统都是用所谓的双曲型微分方程来建模的。例如，用于天气预报、飞机设计以及洋流和生物流体研究的模型都是双曲型微分方程式。这项研究项目涉及到双曲型微分方程解的逼近技术的发展。更确切地说，该项目继续开发具有非光滑解的双曲型方程的数值方法(例如，当模型本身中出现的项受到非光滑的影响时；例如，由于非光滑的边界或界面)。在这种情况下，有效的方法必须能够去除由非光滑性引入的近似伪影，同时尽可能地保持接近潜在预期真实解的行为。该项目对科学和工程应用领域的影响将在于引入技术，以确保基于数值方法的更准确和更有效的解决方案。其他更广泛的影响包括指导和与研究生合作，以及通过研讨会和公开讲座向更大的科学界传播研究成果。该项目研究双曲型微分方程的数值方法，特别是对于没有光滑解的问题，其中准确的计算在很大程度上依赖于非线性极限方法。PI将开发几种技术来改进现有的数值方法，并开发新的方法。该项目的目标之一是为“往返误差补偿和校正方法”开发新的限制方法。该项目的第二个目标是改进双曲型守恒律的Runge-Kutta间断Galerkin方法的Courant-Friedrichs-Lewis(CFL)数。这里，PI和合作者的主要思想是使用额外的守恒约束作为对变分能量泛函的惩罚，从而在不增加复杂性或降低精度阶数的情况下获得数倍大的CFL数。

Many natural systems are modeled by differential equations of so-called hyperbolic type. For example, models used for weather forecasting, aircraft design, and study of ocean currents and biofluids are all hyperbolic differential equations. This research project concerns the development of techniques to approximate the solutions of hyperbolic differential equations. More precisely, the project continues the development of numerical methods for hyperbolic equations with non-smooth solutions (such as is the case when the terms appearing in the model itself are subject to the influence of non-smoothness; say, because of non-smooth boundaries, or interfaces). In such cases, effective methods must be able to remove approximation artifacts introduced by the non-smoothness, while maintaining as much as possible a behavior close to that of the underlying expected true solution. The impact of the project on applied domains in the sciences and engineering will be in introducing techniques to guarantee more accurate and efficient solutions based on numerical methods. Additional broader impacts include mentoring and collaborating with a graduate student and disseminating the research findings to the larger scientific community through seminars and public lectures. The project studies numerical methods for hyperbolic differential equations, in particular for problems that do not have smooth solutions, where accurate computations largely depend on nonlinear limiting methods. The PI will develop several techniques to improve existing numerical methods and also to develop new methods. One goal of the project is the development of new limiting methods for the "back and forth error compensation and correction method." A second goal of the project is to improve on the Courant-Friedrichs-Lewy (CFL) numbers of Runge-Kutta discontinuous Galerkin methods for hyperbolic conservation laws. Here, the main idea of the PI and collaborators is to use extra conservation constraints as penalty for the variational energy functional, and thereby achieve CFL numbers several times larger without increasing the complexity or reducing order of accuracy.