In this paper, we propose an efficient, high order accurate and asymptotic-preserving (AP) semi-Lagrangian (SL) method for the BGK model with constant or spatially dependent Knudsen number. The spatial discretization is performed by a mass conservative nodal discontinuous Galerkin (NDG) method, while the temporal discretization of the stiff relaxation term is realized by stiffly accurate diagonally implicit Runge-Kutta (DIRK) methods along characteristics. Extra order conditions are enforced for asymptotic accuracy (AA) property of DIRK methods when they are coupled with a semi-Lagrangian algorithm in solving the BGK model. A local maximum principle preserving (LMPP) limiter is added to control numerical oscillations in the transport step. Thanks to the SL and implicit nature of time discretization, the time stepping constraint is relaxed and it is much larger than that from an Eulerian framework with explicit treatment of the source term. Extensive numerical tests are presented to verify the high order AA, efficiency and shock capturing properties of the proposed schemes.

中文翻译：

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BGK模型的半拉格朗日节点不连续Galerkin方法

在本文中，我们为具有恒定或空间相关Knudsen数的BGK模型提出了一种高效，高阶准确且渐近保留（AP）的半拉格朗日（SL）方法。空间离散化是通过质量保守节点不连续伽勒金（NDG）方法执行的，而刚性松弛项的时间离散化则是通过沿特征的刚性精确对角隐式Runge-Kutta（DIRK）方法实现的。当DIRK方法与半拉格朗日算法结合用于求解BGK模型时，将为DIRK方法的渐进精度（AA）属性强制执行额外的阶数条件。添加了局部最大原理保留（LMPP）限制器，以控制传输步骤中的数值振荡。得益于SL和时间离散化的隐性，时间步进约束得到了放宽，它比带有显式处理源项的欧拉框架的约束大得多。提出了广泛的数值测试，以验证所提出方案的高阶AA，效率和震动捕获特性。