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.
