We consider dynamical low-rank approximation (DLRA) for the numerical simulation of Vlasov--Poisson equations 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. We propose a variational formulation of the projector splitting which allows to handle inflow boundary conditions on spatial domains with piecewise linear boundary. Numerical experiments demonstrate the principle feasibility of this approach.

翻译：我们考虑基于空间与速度变量分离的Vlasov-Poisson方程数值模拟中的动态低秩逼近方法，该思路已在近期多项研究中被提出。在动态低秩逼近模型的时间积分中，标准方法依据分离变量对低秩流形的切空间投影算子进行分裂，并可通过修正实现秩自适应。然而，边界条件在动态低秩逼近模型中的融入问题尚未得到充分研究。本文提出一种投影分裂的变分形式，可在具有分段线性边界的空间域上处理入流边界条件。数值实验验证了该方法的原理可行性。