A low rank tensor representation of linear transport and nonlinear Vlasov solutions and their associated flow maps

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