A Low Rank Tensor Representation of Linear Transport and Nonlinear Vlasov Solutions and Their Associated Flow Maps

Abstract

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.

References in corpus (3)

• A mass, momentum, and energy conservative dynamical low-rank scheme for the Vlasov equation
• Dynamic tensor approximation of high-dimensional nonlinear PDEs
• An efficient dynamical low-rank algorithm for the Boltzmann-BGK equation close to the compressible viscous flow regime

Cited by in corpus (5)

• A robust and conservative dynamical low-rank algorithm
• Efficient 6D Vlasov simulation using the dynamical low-rank framework Ensign
• Dynamical low-rank approximation of the Vlasov-Poisson equation with piecewise linear spatial boundary
• Sylvester-Preconditioned Adaptive-Rank Implicit Time Integrators for Advection-Diffusion Equations with Variable Coefficients
• Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations