TY - JOUR A1 - Uschmajew, André A1 - Zeiser, Andreas T1 - Dynamical low-rank approximation of the Vlasov–Poisson equation with piecewise linear spatial boundary JF - BIT Numerical Mathematics N2 - Dynamical low-rank approximation (DLRA) for the numerical simulation of Vlasov–Poisson equations is based on separation of space and velocity variables, as proposed in several recent works. The standard approach for the time integration in the DLRA model uses a splitting of the tangent space projector for the low-rank manifold according to the separated variables. It can also be modified to allow for rank-adaptivity. A less studied aspect is the incorporation of boundary conditions in the DLRA model. In this work, a variational formulation of the projector splitting is proposed which allows to handle inflow boundary conditions on spatial domains with piecewise linear boundary. Numerical experiments demonstrate the principle feasibility of this approach. KW - Dynamical low-rank approximation KW - Boundary condition KW - Projector-splitting integrator KW - Friedrichs’ system KW - 15A69 KW - 65M60 KW - 82D10 Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:523-19786 SN - 0006-3835 SN - 1572-9125 VL - 64 IS - 2 PB - Springer Nature CY - Dordrecht ER - TY - JOUR A1 - Uschmajew, André A1 - Zeiser, Andreas T1 - Discontinuous Galerkin discretization of conservative dynamical low-rank approximation schemes for the Vlasov–Poisson equation JF - BIT Numerical Mathematics N2 - A numerical dynamical low-rank approximation (DLRA) scheme for the solution of the Vlasov–Poisson equation is presented. Based on the formulation of the DLRA equations as Friedrichs’ systems in a continuous setting, it combines recently proposed conservative DLRA methods with a discontinuous Galerkin discretization. The resulting scheme is shown to ensure mass and momentum conservation at the discrete level. In addition, a new formulation of the conservative integrator is proposed based on its interpretation as a tangent space projector splitting scheme. Numerical experiments validate our approach in one- and two-dimensional simulations of Landau damping. As a demonstration of feasibility, it is also shown that the rank-adaptive unconventional integrator can be combined with mesh adaptivity. KW - - KW - Dynamical low-rank approximation KW - Discontinuous Galerkin method KW - Friedrichs’ system KW - Mass and momentum conservation KW - 15A69 KW - 65M60 KW - 82D10 Y1 - 2025 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:523-21753 SN - 0006-3835 SN - 1572-9125 VL - 65 IS - 4 PB - Springer Nature CY - Dordrecht ER - TY - JOUR A1 - Uschmajew, André A1 - Zeiser, Andreas T1 - Discontinuous Galerkin discretization of conservative dynamical low-rank approximation schemes for the Vlasov–Poisson equation JF - BIT Numerical Mathematics N2 - A numerical dynamical low-rank approximation (DLRA) scheme for the solution of the Vlasov–Poisson equation is presented. Based on the formulation of the DLRA equations as Friedrichs’ systems in a continuous setting, it combines recently proposed conservative DLRA methods with a discontinuous Galerkin discretization. The resulting scheme is shown to ensure mass and momentum conservation at the discrete level. In addition, a new formulation of the conservative integrator is proposed based on its interpretation as a tangent space projector splitting scheme. Numerical experiments validate our approach in one- and two-dimensional simulations of Landau damping. As a demonstration of feasibility, it is also shown that the rank-adaptive unconventional integrator can be combined with mesh adaptivity. KW - Dynamical low-rank approximation KW - Discontinuous Galerkin method KW - Friedrichs’ system KW - Mass and momentum conservation KW - 15A69 KW - 65M60 KW - 82D10 Y1 - 2025 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:523-21983 SN - 0006-3835 SN - 1572-9125 VL - 65 IS - 4 PB - Springer CY - Dordrecht ER -