Multiscale Modeling &Simulation, Volume 19, Issue 1, Page 568-587, January 2021.
The Fokker--Planck approximation for an elementary linear, two-dimensional kinetic model endowed with a mass-preserving integral collision process is numerically studied, along with its diffusive limit. In order to set up a well-balanced discretization relying on an $S$-matrix, exact steady states of the continuous equation are derived. The ability of the scheme to keep these stationary solutions invariant produces the discretization of the local differential operator which mimics the collision process. The aforementioned scheme can be reformulated as an implicit-explicit one, which is proved to be both well-balanced and asymptotic-preserving in the diffusion limit. Several numerical benchmarks, computed on coarse grids, are displayed so as to illustrate the results.

中文翻译：

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自由克莱因-克莱默方程的二维均衡格式的扩散极限

多尺度建模与仿真，第 19 卷，第 1 期，第 568-587 页，2021 年 1 月。
对具有质量保持积分碰撞过程的基本线性二维动力学模型的 Fokker-Planck 近似及其扩散极限进行了数值研究。为了建立依赖于 $S$ 矩阵的良好平衡的离散化，需要推导出连续方程的精确稳态。该方案保持这些固定解不变的能力产生了模拟碰撞过程的局部微分算子的离散化。上述方案可以重新表述为隐式-显式方案，它被证明在扩散极限中既是平衡的又是渐近保持的。显示了在粗网格上计算的几个数值基准，以说明结果。