Development of Discontinuous Galerkin Methods for Kinetic Transport Models and Control Problems with State Constraints

动态输运模型和状态约束控制问题的不连续伽辽金方法的发展

• 批准号: 1217563
• 负责人: Yingda Cheng
• 金额: $ 7.33万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-11-10 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1217563&HistoricalAwards=false
• 关键词: Development; Discontinuous; Galerkin; Methods; Kinetic

The objective of this project is to develop and analyze novel discontinuous Galerkin (DG) methods for solving partial differential equations arising from various application areas. The DG method is a class of finite element methods using completely discontinuous piecewise polynomial space for the numerical solution and the test functions. Those robust, compact, locally conservative methods can treat arbitrarily unstructured meshes and are ideal for hp-adaptive strategies. The good properties of the scheme call for further research in areas that are traditionally not solved by DG methods. In this grant proposal, the PI plans to conduct research in the following directions: (1) a positivity-preserving DG method for solving the kinetic equations, including the Boltzmann equations and Vlasov equations, (2) application of the proposed method to solar cell/semiconductor device simulations and plasma physics, (3) a novel DG solver for the Hamilton-Jacobi equations and its applications in control problems with state constraints. The proposed activity lies between algorithm development, analysis and applications. Developing robust, high-order accurate, cost-efficient numerical algorithms for kinetic models and control problem is very challenging, not only because of the high dimensionality of such models, but also because of the fact that a deep understanding of the underlying physics is required. The eventual goal is to produce solvers that are computationally efficient and suit the need for applications. The PI's work arises from the computational demand of real world applications. Many ideas developed in this proposal will have straightforward applications and impacts in semiconductor device simulations, high-efficiency fuel cell modeling, control problems and plasma physics. The PI actively interacts with students and faculty members in mathematics, physics, electrical engineering and chemistry departments. In addition, the PI will integrate the project with the training of graduate students in order to communicate in a broader context.

该项目的目的是开发和分析新型的不连续的盖尔金（DG）方法，以解决由各种应用领域引起的部分微分方程。 DG方法是使用完全不连续的分段多项式空间用于数值解决方案和测试函数的一类有限元方法。那些坚固，紧凑，当地保守的方法可以治疗任意非结构化的网格，并且是HP自适应策略的理想选择。该计划的良好特性要求在传统上无法通过DG方法解决的领域进行进一步研究。在这项赠款提案中，PI计划在以下方向上进行研究：（1）一种阳性的DG DG方法，用于求解动力学方程，包括玻尔兹曼方程和弗拉索夫方程，（2）应用于太阳能电池/半导体设备仿真和dg sol的应用中，将所提出的方法应用于Solar/Semicodoctor solics，（3）状态约束。 所提出的活动在于算法开发，分析和应用之间。 为动力学模型和控制问题开发可靠的，高阶的准确，成本效益的数值算法是非常具有挑战性的，不仅是因为此类模型的高维度很高，而且还因为对基础物理学的深入了解。最终的目标是生产计算上有效的求解器，并适合应用程序的需求。 PI的工作来自现实世界应用的计算需求。该提案中提出的许多想法将在半导体设备模拟，高效燃料电池建模，控制问题和等离子体物理学中具有直接的应用和影响。 PI与数学，物理，电气工程和化学部门的学生和教职员工积​​极互动。 此外，PI将将项目与研究生的培训相结合，以便在更广泛的背景下进行交流。