Numerical Discretisation of Hyperbolic Systems of Moment Equations Describing Sedimentation in Suspensions of Rod-Like Particles

TL;DR

The paper develops adaptive, hyperbolic moment-closure methods for sedimentation models of rod-like particles, combining kinetic descriptions with fluid dynamics. It introduces a splitting-based numerical framework and a conservative high-resolution Wave Propagation Algorithm that supports local changes in moment resolution via interface coupling, guided by a theoretically justified error indicator. An error-estimation analysis based on a generalized Gronwall lemma bounds the difference between $2N+1$ and $2N+3$ moment models, enabling reliable adaptivity. The methods are demonstrated in 1D and 2D shear and flow settings, showing second-order convergence and effective local refinement, with bulk-coupling to flow equations and robust handling of interfaces between differing moment counts. This yields efficient simulations of concentration instabilities and cluster formation in sedimenting suspensions while preserving conservation and hyperbolic structure.

Abstract

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.

Figures (6)

• Figure 1: Structure of the solution $Q$ of the generalised Riemann Problem (\ref{['eqn:generaliesedRP']}) for $N=1$ and $M=2$. The value of $Q$ is constant in each wedge of the $x-t$ plane.
• Figure 2: Solution $\rho$ of the generalised Riemann problem (\ref{['eqn:generaliesedRP']}) described in \ref{['exa:Interface1']} at time $t=5$. For $x<0$ we used $w_x/D_r=1$, for $x>0$ we used $w_x/D_r=4$. The blue dashed dotted curve is the solution for $N=1$ and $M=2$. The red dashed curve uses $N=M=1$ and the green dotted curve uses $N=M=2$ moment equations. The black solid curve is a highly resolved reference solution.
• Figure 3: Solution $\rho$ of the generalised Riemann problem described in \ref{['exa:Interface1']} at time $t=5$. For $x<0$ we use $w_x/D_r=1$, for $x>0$ we use $w_x/D_r=4$. The blue dashed dotted curve uses different values of $N$ on the left hand side of the interface and $M=20$ on the right hand side of the interface. This solution is compared with a rough solution (red dashed curve) and a detailed solution (green dotted curve) which use $2N+1=2M+1$ moment equations throughout the domain. The black solid curve is a highly resolved reference solution.
• Figure 4: Solution $\rho$ of the generalised Riemann problem described in \ref{['exa:Interface1']} at time $t=5$. For $x<0$ we used $w_x/D_r=10$, for $x>0$ we used $w_x/D_r=40$. The blue dashed dotted curve uses different values of $N$ on the left hand side of the interface and $M=20$ on the right hand side of the interface. This solution is compared with a rough solution (red dashed curve) and a detailed solution (green dotted curve) which use $2N+1=2M+1$ moment equations throughout the domain. The black solid curve is a highly resolved reference solution.
• Figure 5: Schematic diagram of Godunov's method for solving the generalised Riemann Problem (\ref{['eqn:generaliesedRP']}) for $N=1$ and $M=2$. The Riemann problem is solved at each cell interface.

Theorems & Definitions (8)

• Lemma 3.1
• Proposition 3.2
• Theorem 3.3
• proof
• Example 4.3
• Theorem 4.4
• proof
• Example 5.2