A stable rank-adaptive step-and-truncate finite volume method for Vlasov transport on domains with piecewise linear boundaries
Abstract
We consider the numerical solution of the linear Vlasov transport equation on bounded spatial domains with inflow boundary conditions based on low-rank approximation. We combine a finite volume discretization with a rank-adaptive step-and-truncate scheme for the resulting matrix ODE. The spatial and velocity meshes may be unstructured, while suitable numerical fluxes retain a separated space-velocity representation. For homogeneous inflow, we show that the low-rank scheme inherits the L2 stability and CFL restriction of the underlying full finite volume forward Euler method. In addition, the low-rank approximation error is bounded explicitly in terms of the truncation tolerances, avoiding the modeling error associated with tangent-space projections in dynamical low-rank approximation. Numerical experiments in 1d1v and 2d2v confirm the predicted error behavior. In 2d2v, the method handles an unstructured spatial mesh with nonzero inflow and a full tensor-product discretization of approximately 5.8·1010 phase-space cells while the numerical rank is at most twelve.
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BibTeX
@misc{uschmajew2026stable,
title = {{A stable rank-adaptive step-and-truncate finite volume method for Vlasov transport on domains with piecewise linear boundaries}},
author = {André Uschmajew and Andreas Zeiser},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02721},
eprint = {2609.02721},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02721},
}
RIS
TY - UNPB
TI - A stable rank-adaptive step-and-truncate finite volume method for Vlasov transport on domains with piecewise linear boundaries
AU - André Uschmajew
AU - Andreas Zeiser
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02721
UR - https://arxiv.org/abs/2609.02721
AB - We consider the numerical solution of the linear Vlasov transport equation on bounded spatial domains with inflow boundary conditions based on low-rank approximation. We combine a finite volume discretization with a rank-adaptive step-and-truncate scheme for the resulting matrix ODE. The spatial and velocity meshes may be unstructured, while suitable numerical fluxes retain a separated space-velocity representation. For homogeneous inflow, we show that the low-rank scheme inherits the L_2 stability and CFL restriction of the underlying full finite volume forward Euler method. In addition, the low-rank approximation error is bounded explicitly in terms of the truncation tolerances, avoiding the modeling error associated with tangent-space projections in dynamical low-rank approximation. Numerical experiments in 1d1v and 2d2v confirm the predicted error behavior. In 2d2v, the method handles an unstructured spatial mesh with nonzero inflow and a full tensor-product discretization of approximately 5.8·10^10 phase-space cells while the numerical rank is at most twelve.
ER -