CAREER: Development of Discontinuous Galerkin Methods for Kinetic Equations in High Dimensions

职业：高维动力学方程不连续伽辽金方法的发展

• 批准号: 1453661
• 负责人: Yingda Cheng
• 金额: $ 40万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1453661&HistoricalAwards=false
• 关键词: CAREER; Development; Discontinuous; Galerkin; Methods

Kinetic equations arise as fundamental models in many applications such as rarefied gas dynamics, plasma physics, nuclear engineering, semiconductor device design, traffic networking, and swarming. Due to the high-dimensionality of such models, conventional numerical partial differential equation (PDE) solvers will incur prohibitive computational cost, limiting their applications to real-world problems. This research project aims at addressing this issue by developing discontinuous Galerkin (DG) methods by the sparse grid approach. The resulting schemes enjoy the excellent properties that the traditional DG methods offer, while still being able to achieve high-order accuracy with a significant reduction in the required degrees of freedom. The research will have direct impact on the efficient and robust computations of kinetic equations and on other application areas involving high-dimensional PDEs. The research plan is complemented by educational and outreach activities involving the training of undergraduates, graduate students, and postdoctoral associates, and fostering collaborations among female researchers in computational mathematics. The research plan consists of several coherent projects, ranging from algorithm design, analysis, and implementation to application. In particular, the investigator plans to perform detailed studies of sparse tensor product polynomial spaces and to construct DG methods on sparse grids for model equations. The methods will be further developed for kinetic equations with attention to numerical challenges specific to Vlasov and Boltzmann equations. The PI will also develop similar numerical schemes for PDEs arising from areas such as optimal control and mathematical finance. Theoretical issues including stability, conservation, and error estimates, and computational issues including efficient linear solvers, adaptivity, and parallel implementations will be explored. The educational goals of this project are to develop a pipeline for the recruitment, retention, and early research exposure of undergraduate students, to continue integrated training and mentoring of graduate students and postdocs, and to promote collaborations and build a network of support for early career female researchers. Undergraduates, graduate students, and postdocs will be directly involved in every aspect of the proposed research. Summer research and career development workshops at Michigan State University will be established to promote research collaborations and build a community of support for early career female researchers.

在稀薄气体动力学、等离子体物理、核工程、半导体器件设计、交通网络和蜂群等许多应用中，动力学方程都是基本模型。由于这些模型的高维性，传统的数值偏微分方程（PDE）求解方法将产生令人望而却步的计算成本，限制了它们在现实世界问题中的应用。本研究项目旨在通过稀疏网格方法开发不连续伽辽金（DG）方法来解决这一问题。由此产生的方案享有传统DG方法提供的优良特性，同时仍然能够实现高阶精度，所需自由度显着降低。该研究将对动力学方程的高效鲁棒计算以及其他涉及高维偏微分方程的应用领域产生直接影响。该研究计划还包括教育和推广活动，包括对本科生、研究生和博士后的培训，以及促进女性研究人员在计算数学方面的合作。研究计划由几个连贯的项目组成，从算法设计、分析、实现到应用。特别是，研究者计划对稀疏张量积多项式空间进行详细的研究，并在模型方程的稀疏网格上构建DG方法。该方法将进一步发展动力学方程，并注意Vlasov和Boltzmann方程特有的数值挑战。PI还将为最优控制和数学金融等领域产生的pde制定类似的数值方案。理论问题包括稳定性、守恒和误差估计，计算问题包括有效的线性求解器、自适应和并行实现将被探索。该项目的教育目标是建立一个招收、保留和早期研究本科生的渠道，继续对研究生和博士后进行综合培训和指导，并促进合作，建立一个支持早期职业女性研究人员的网络。本科生、研究生和博士后将直接参与拟议研究的各个方面。密歇根州立大学将设立夏季研究和职业发展研讨会，以促进研究合作，并建立一个支持早期职业女性研究人员的社区。