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

动力学方程在许多应用中作为基本模型出现，如稀薄气体动力学、等离子体物理、核工程、半导体器件设计、交通网络和蜂群。由于这类模型的高维性，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。本研究旨在以稀疏网格法为基础，发展不连续伽辽金(DG)方法来解决这一问题。所得到的格式具有传统DG方法所提供的优良特性，同时仍然能够在显著减少所需自由度的情况下实现高阶精度。这项研究将对动力学方程的高效和稳健的计算以及涉及高维偏微分方程组的其他应用领域产生直接的影响。该研究计划还得到了教育和推广活动的补充，这些活动涉及对本科生、研究生和博士后助理进行培训，并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，并在稀疏网格上构造求解模型方程的DG方法。将进一步发展动力学方程的方法，并注意到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偏微分方程组开发类似的数值格式。理论问题，包括稳定性、守恒性和误差估计，以及计算问题，包括有效的线性求解器、适应性和并行实现，将被探索。该项目的教育目标是为本科生的招聘、保留和早期研究暴露建立一个渠道，继续对研究生和博士后进行综合培训和指导，促进合作并为早期职业女性研究人员建立支持网络。本科生、研究生和博士后将直接参与拟议研究的各个方面。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，并建立一个支持早期职业女性研究人员的社区。

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.