We apply the collision-based hybrid introduced in \cite{hauck} to the Boltzmann equation with the BGK operator and a hyperbolic scaling. An implicit treatment of the source term is used to handle stiffness associated with the BGK operator. Although it helps the numerical scheme become stable with a large time step size, it is still not obvious to achieve the desired order of accuracy due to the relationship between the size of the spatial cell and the mean free path. Without asymptotic preserving property, a very restricted grid size is required to resolve the mean free path, which is not practical. Our approaches are based on the noncollision-collision decomposition of the BGK equation. We introduce the arbitrary order of nodal discontinuous Galerkin (DG) discretization in space with a semi-implicit time-stepping method; we employ the backward Euler time integration for the uncollided equation and the 2nd order predictor-corrector scheme for the collided equation, i.e., both source terms in uncollided and collided equations are treated implicitly and only streaming term in the collided equation is solved explicitly. This improves the computational efficiency without the complexity of the numerical implementation. Numerical results are presented for various Knudsen numbers to present the effectiveness and accuracy of our hybrid method. Also, we compare the solutions of the hybrid and non-hybrid schemes.

翻译：我们应用\cite{hauck}中提出的基于碰撞的混合方法，对具有BGK算子和双曲尺度变换的玻尔兹曼方程进行求解。为处理BGK算子带来的刚性，采用隐式方法处理源项。尽管该方法有助于数值格式在大时间步长下保持稳定，但由于空间网格尺寸与平均自由程之间的关系，仍难以达到期望的精度阶数。若缺乏渐近保持性质，则需要极其严格的网格尺寸来解析平均自由程，这在实践中并不可行。我们的方法基于BGK方程的非碰撞-碰撞分解。在空间离散中引入任意阶节点间断伽辽金(DG)方案，并采用半隐式时间推进方法：对未碰撞方程采用向后欧拉时间积分，对碰撞方程采用二阶预测-校正格式，即未碰撞方程和碰撞方程中的源项均隐式处理，仅碰撞方程中的输运项显式求解。这种方法在不增加数值实现复杂度的前提下提升了计算效率。通过不同克努森数下的数值结果验证了混合方法的有效性与精度，并比较了混合方案与非混合方案的解。