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.

动力学方程作为许多应用中的基本模型出现，例如稀薄气体动力学、等离子体物理、核工程、半导体器件设计、交通网络和集群。 由于此类模型的高维性，传统的数值偏微分方程（PDE）求解器将产生高昂的计算成本，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）方法来解决这个问题。由此产生的方案具有传统 DG 方法所提供的优异特性，同时仍然能够实现高阶精度，同时显着降低所需的自由度。该研究将对动力学方程的高效和鲁棒计算以及涉及高维偏微分方程的其他应用领域产生直接影响。该研究计划还辅以教育和外展活动，包括对本科生、研究生和博士后的培训，以及促进计算数学领域女性研究人员之间的合作。 该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，研究人员计划对稀疏张量积多项式空间进行详细研究，并在稀疏网格上构建模型方程的 DG 方法。这些方法将进一步开发用于动力学方程，并关注 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的偏微分方程开发类似的数值方案。将探讨包括稳定性、守恒和误差估计在内的理论问题，以及包括高效线性求解器、自适应性和并行实现在内的计算问题。该项目的教育目标是为本科生的招募、保留和早期研究接触开发一条渠道，继续对研究生和博士后进行综合培训和指导，并促进合作并建立早期职业支持网络女性研究人员。本科生、研究生和博士后将直接参与拟议研究的各个方面。将在密歇根州立大学设立夏季研究和职业发展研讨会，以促进研究合作并为早期职业女性研究人员建立一个支持社区。