论文标题
Vlasov-ampère-Fokker-Planck系统的有效动力较低近似
Efficient dynamical low-rank approximation for the Vlasov-Ampère-Fokker-Planck system
论文作者
论文摘要
动力学方程由于其高维度而难以在数值上求解。降低计算成本的一种有希望的方法是动态低级算法，该算法通过提出ANSATZ作为位置和速度的可分离（Rank-1）函数的总和来解除相空间的尺寸。在小规模数字限制中获得的碰撞动力学方程式的流体渐近极限，当写入$ f = mg $时，$ m $是本地的麦克斯韦人，而$ g $是低率的，则承认低级表示。我们将这种分解应用于血浆动力学的Vlasov-Ampère-Fokker-Planck方程，考虑到强碰撞和电场的渐近极限。我们实施了我们提出的算法，并通过与进化完整的解决方案张量$ f $的实施相比，证明了计算时间的预期改进。我们还证明，我们的算法可以在动力学方面和流体方案中以相对较低的计算努力捕获动力学，从而有效地捕获了渐近流体的极限。
Kinetic equations are difficult to solve numerically due to their high dimensionality. A promising approach for reducing computational cost is the dynamical low-rank algorithm, which decouples the dimensions of the phase space by proposing an ansatz as the sum of separable (rank-1) functions in position and velocity respectively. The fluid asymptotic limit of collisional kinetic equations, obtained in the small-Knudsen number limit, admits a low-rank representation when written as $f = Mg$, where $M$ is the local Maxwellian, and $g$ is low-rank. We apply this decomposition to the Vlasov-Ampère-Fokker-Planck equation of plasma dynamics, considering the asymptotic limit of strong collisions and electric field. We implement our proposed algorithm and demonstrate the expected improvement in computation time by comparison to an implementation that evolves the full solution tensor $f$. We also demonstrate that our algorithm can capture dynamics in both the kinetic regime, and in the fluid regime with relatively lower computational effort, thereby efficiently capturing the asymptotic fluid limit.
