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Biosketch: Professor Jingmei Qiu got her B.S. degree from university of science and technology of China in 2003; she got her Ph.D. from Brown University under the supervision of Professor Chi-Wang Shu in 2007. She spent a year at Michigan State University working with Professor Andrew Christlieb from 2007 to 2008. She held tenure track faculty position in Colorado School of Mines 2008-2011, in University of Houston from 2011 to 2017 and was promoted to Associate Professor in 2014. She moved to University of Delaware in 2017 and was promoted to Professor in 2019. 

Professor Qiu’s research interests include high order numerical methods for fluid, kinetic and multi-scale models. Recently she is interested in Eulerian-Lagrangian high order approaches and low rank approximations to high dimensional nonlinear dynamics. 

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.