We present a numerical discretisation of the coupled moment systems, previously introduced in Dahm and Helzel, which approximate the kinetic multi-scale model by Helzel and Tzavaras for sedimentation in suspensions of rod-like particles for a two-dimensional flow problem and a shear flow problem. We use a splitting ansatz which, during each time step, separately computes the update of the macroscopic flow equation and of the moment system. The proof of the hyperbolicity of the moment systems in \cite{Dahm} suggests solving the moment systems with standard numerical methods for hyperbolic problems, like LeVeque's Wave Propagation Algorithm \cite{LeV}. The number of moment equations used in the hyperbolic moment system can be adapted to locally varying flow features. An error analysis is proposed, which compares the approximation with $2N+1$ moment equations to an approximation with $2N+3$ moment equations. This analysis suggests an error indicator which can be computed from the numerical approximation of the moment system with $2N+1$ moment equations. In order to use moment approximations with a different number of moment equations in different parts of the computational domain, we consider an interface coupling of moment systems with different resolution. Finally, we derive a conservative high-resolution Wave Propagation Algorithm for solving moment systems with different numbers of moment equations.

翻译：我们提出了一种耦合矩系统的数值离散化方法，该方法此前由Dahm和Helzel提出，用于近似Helzel和Tzavaras提出的描述棒状颗粒悬浮沉降的动力学多尺度模型，适用于二维流动问题和剪切流动问题。我们采用分裂策略，在每个时间步内分别计算宏观流动方程和矩系统的更新。文献\cite{Dahm}中关于矩系统双曲性的证明表明，可采用标准双曲问题数值方法（如LeVeque的波动传播算法\cite{LeV}）进行求解。双曲矩系统中使用的矩方程数量可根据局部流动特征自适应调整。我们提出了一种误差分析方法，将包含$2N+1$个矩方程的近似解与包含$2N+3$个矩方程的近似解进行对比。该分析推导出一个误差指示器，该指示器可直接从包含$2N+1$个矩方程的数值近似解计算得出。为在计算域不同区域使用不同数量的矩方程，我们考虑了不同分辨率矩系统间的界面耦合。最后，我们推导出一种守恒型高分辨率波动传播算法，用于求解不同矩方程数量的矩系统。