Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our decomposition is compatible with the projector-splitting integrator, and can therefore access second-order accuracy in time via a Strang splitting scheme. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as a second-order accurate integrator which exactly conserves charge and energy. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems.

翻译：动态低秩（DLR）近似近年来作为解决Boltzmann和Vlasov等动力学方程数值求解中维度灾难的有效方案受到关注。这类方法包括投影分裂和基更新-伽辽金DLR积分器，在显著提升动力学求解的计算效率方面展现出潜力。然而，这往往以牺牲电荷、电流和能量的守恒性为代价。本文展示了一种新颖的宏观-微观分解，可将分布函数分离为两个分量：其中一个分量承载守恒量，另一个分量与之正交。我们将DLR近似应用于后者，从而获得一种简洁且可扩展的守恒DLR方案，该方案保留了基础方案的计算优势。此外，我们的分解与投影分裂积分器兼容，因此可通过Strang分裂格式实现时间二阶精度。我们描述了一种能够精确守恒电荷及电流或能量的一阶积分器，以及一种精确守恒电荷和能量的二阶精度积分器。为突显所提出宏观-微观分解的灵活性，我们实现了两种速度空间离散化方法，并通过一组等离子体基准问题验证了所声称的精度与守恒特性。