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 (BUG) 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 approach requires no change to the mechanics of the DLR approximation, so it is compatible with both the BUG family of integrators and the projector-splitting integrator which we use here. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as an integrator which exactly conserves charge and energy and exhibits second-order accuracy on our test problems. 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数灾难问题的可行方案而受到关注。这类方法包括投影分裂法和基更新-伽辽金（BUG）DLR积分器，在显著提升动力学求解计算效率方面展现出应用前景。然而，这通常以牺牲电荷、电流和能量守恒为代价。本研究展示如何通过新型宏观-微观分解将分布函数分离为两个分量：其中一个分量承载守恒量，另一分量与之正交。我们对后者应用DLR近似，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，本方法无需改变DLR近似的实现机制，因此既兼容我们此处使用的BUG族积分器，也兼容投影分裂积分器。我们描述了一阶积分器（可精确守恒电荷与电流或能量）以及二阶积分器（在测试问题中精确守恒电荷与能量并展现二阶精度）。为突出所提出的宏观-微观分解的灵活性，我们实现了两种速度空间离散化方案，并在系列等离子体基准问题上验证了所声称的精度和守恒特性。