Abstract:
In this talk, we introduce a sampling-based semi-Lagrangian adaptive rank (SLAR) method, which leverages a cross-approximation strategy - or pseudo-skeleton decomposition - to efficiently represent low-rank structures in kinetic solutions. The method dynamically adapts the rank of the solution while ensuring numerical stability through singular value truncation and mass-conservative projections. By combining the advantages of semi-Lagrangian integration with low-rank approximations, SLAR enables significantly larger time steps compared to conventional methods and is extended to nonlinear systems such as the Vlasov-Poisson equations using a Runge-Kutta exponential integrator.
Building on this framework, we further develop the SLAR method for the multi-scale Bhatnagar-Gross-Krook equation, introducing an asymptotically accurate approach that eliminates the need for low-rank decompositions of the local Maxwellian in the collision operator. To enforce conservation of mass, momentum, and energy, we propose a novel locally macroscopic conservative technique, which discretizes the macroscopic system using high-order diagonally-implicit Runge–Kutta methods. Additionally, a dynamic closure strategy is employed to self-consistently adjust macroscopic moments, enabling robust simulations across both kinetic and hydrodynamic regimes, even in the presence of shocks and discontinuities.
Speaker’s Bio:
Dr. Jingmei Qiu is an Unidel Professor in the Department of Mathematical Sciences at the University of Delaware. Her research focuses on the design, analysis, and application of high-order structure-preserving computational algorithms for complex systems characterized by multi-scale, multi-physics, and high-dimensional features. Dr. Qiu’s work includes developing low-rank tensor approximations for high-dimensional, time-dependent problems with structure preservation, as well as Eulerian-Lagrangian high-order numerical methods for fluid and kinetic applications. She was awarded the Air Force Young Investigator Award in 2012 and is the lead Principal Investigator of a Multidisciplinary University Research Initiative project on Tensor Networks, supported by the Department of Defense from 2024 to 2029.
