  We propose a low-rank tensor approach to approximate linear transport and
nonlinear Vlasov solutions and their associated flow maps. The approach takes
advantage of the fact that the differential operators in the Vlasov equation is
tensor friendly, based on which we propose a novel way 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. We apply the same procedure to evolve the
dynamics of the flow map in a low-rank fashion, which proves to be advantageous
when the flow map enjoys the low rank structure, while the solution suffers
from high rank or displays filamentation structures. Hierarchical Tucker
decomposition is adopted for high dimensional problems. An extensive set of
linear and nonlinear Vlasov test examples are performed to show the high order
spatial and temporal convergence of the algorithm with mesh refinement up to
SVD-type truncation, the significant computational savings of the proposed
low-rank approach especially for high dimensional problems, the improved
performance of the flow map approach for solutions with filamentations.

Numerical Analysis;Numerical Analysis