Speaker
Dr. Jingmei Qiu, Professor, Department of Mathematical Sciences, University of Delaware
Title
Mathematics Seminar Series
Subtitle
A Conservative Adaptive Low Rank High Order Tensor Approach for Nonlinear Vlasov Equations
Digital Location
https://msstate.webex.com/msstate/j.php?MTID=ma26e4b5158152e2b1f03263012c24244
Abstract: We propose a conservative adaptive low-rank tensor approach to approximate nonlinear Vlasov solutions. The approach takes advantage of the fact that the differential operators in the Vlasov equation is tensor friendly, based on which we propose to dynamically and adaptively build up low-rank solution basis by adding new basis functions from discretization of the PDE, and removing basis from an SVD-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization and a second order strong stability preserving multi-step time discretization.
While the SVD truncation will destroy the conservation properties of the full rank conservative scheme, we further develop low rank schemes with local mass, momentum and energy conservation for the corresponding macroscopic equations. The mass and momentum conservation are achieved by a conservative SVD truncation, while the energy conservation is achieved by replacing the energy component of the kinetic solution by the ones obtained from conservative schemes for macroscopic energy equation.
Hierarchical Tucker decomposition is adopted for high dimensional problems, overcoming the curse of dimensionality. An extensive set of linear and nonlinear Vlasov examples are performed to show the high order spatial and temporal convergence of the algorithm, the significant CPU and storage savings of the proposed low-rank approach especially for high dimensional problems, as the local conservation of macroscopic mass, momentum and energy.
(Joint work with Wei Guo from Texas Tech University)
Biosketch: Dr. Qiu received her PhD degree in Applied Mathematics from Brown University in 2007 under the supervision of Prof. Chi-Wang Shu. After spending part of her career as Assistant Professor at Colorado School of Mines (2008 – 2011) and University of Houston (2011 – 2017), she joined the University of Delaware as Associate Professor in 2017 and had been promoted to Full Professor since 2019. She was a recipient of Young Investigator Award 2012 from AFOSR and has been a sole PI of numerous research grants from NSF and AFOSR. Her research interests include high order multi-scale numerical methods for kinetic and transport problems.
For more information, please contact: Dr. Vu Thai Luan, luan@math.msstate.edu, (662)-325-7162.