Multiscale Modeling &Simulation, Volume 19, Issue 1, Page 184-207, January 2021.
We investigate the numerical discretization of a two-stream kinetic system with an internal state; such a system has been introduced to model the motion of cells by chemotaxis. This internal state models the intracellular methylation level. It adds a variable in the mathematical model, which makes it more challenging to simulate numerically. Moreover, it has been shown that the macroscopic or mesoscopic quantities computed from this system converge to the Keller--Segel system at diffusive scaling or to the velocity-jump kinetic system for chemotaxis at hyperbolic scaling. Then we propose numerical schemes that are uniformly accurate with respect to the scaling parameter. We show that these schemes converge to some limiting schemes which are consistent with the limiting macroscopic or kinetic system. This study is illustrated with some numerical simulations and comparisons with Monte Carlo simulations.

中文翻译：

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具有内部状态的动力学输运方程的数值方案

多尺度建模与仿真，第 19 卷，第 1 期，第 184-207 页，2021 年 1 月。
我们研究了具有内部状态的两流动力学系统的数值离散化；已经引入这样的系统来模拟细胞趋化性的运动。这种内部状态模拟细胞内甲基化水平。它在数学模型中增加了一个变量，这使得数值模拟更具挑战性。此外，已经表明，从该系统计算出的宏观或细观量在扩散标度下收敛到 Keller--Segel 系统，或在双曲线标度下收敛到趋化性的速度跳跃动力学系统。然后我们提出了关于缩放参数一致准确的数值方案。我们表明这些方案收敛于一些与极限宏观或动力学系统一致的极限方案。