In this work, high order asymptotic preserving schemes are constructed and analysed for kinetic equations under a diffusive scaling. The framework enables to consider different cases: the diffusion equation, the advection-diffusion equation and the presence of inflow boundary conditions. Starting from the micro-macro reformulation of the original kinetic equation, high order time integrators are introduced. This class of numerical schemes enjoys the Asymptotic Preserving (AP) property for arbitrary initial data and degenerates when $\epsilon$ goes to zero into a high order scheme which is implicit for the diffusion term, which makes it free from the usual diffusion stability condition. The space discretization is also discussed and high order methods are also proposed based on classical finite differences schemes. The Asymptotic Preserving property is analysed and numerical results are presented to illustrate the properties of the proposed schemes in different regimes.

翻译：本文构建并分析了扩散尺度下动能方程的高阶渐近保持格式。该框架可处理不同情形：扩散方程、对流-扩散方程以及存在流入边界条件的情况。基于原始动能方程的微观-宏观重构，引入了高阶时间积分器。该类数值方案对任意初始数据均具有渐近保持(AP)性质，当$\epsilon$趋近于零时退化为扩散项隐式的高阶格式，从而免除了通常的扩散稳定性条件。此外还讨论了空间离散化问题，基于经典有限差分方案提出了高阶空间离散方法。文中分析了渐近保持性质，并通过数值结果展示了所提方案在不同区域中的特性。