Micro-Macro Simulation of Shallow Water Moment Equations

TL;DR

The paper develops a micro-macro coupling for the shallow water moment equations (SWME), leveraging a vertical velocity expansion $u = u_m + \sum_{j=1}^N \alpha_j \phi_j(\zeta)$ to form a (N+2)-variable system $U = (h, hu_m, h\alpha_1, \dots, h\alpha_N)^T$. By combining a high-fidelity micro model ($M$ variables) with a lower-order macro model ($L$ variables, $M>L$) through a four-step loop—micro step, restriction, macro step, and matching—the method aims to retain SWME accuracy while enabling larger time steps and reduced computation. The matching step exploits Legendre orthogonality to reconstruct micro states from macro states, while complexity analysis shows per-iteration cost scales as $\mathcal{O}\left(\frac{M^2+L^2+M}{2\Delta x}\right)$, favoring configurations with large $M-L$ and modest $L$. Numerical results on dam-break and wave-transport tests demonstrate substantial speedups (often >$2\times$) with accuracy that improves as the macro model becomes richer, indicating practical potential for adaptive or domain-decomposed implementations in shallow-water simulations.

Abstract

Shallow flows are governed by the Navier-Stokes equations. They are commonly modelled using the shallow water equations, a great simplification of the Navier-Stokes equations, which often yields inaccurate results. For that reason, a model called shallow water moment equations has been developed. It uses more equations and variables than the shallow water equations. While this model is significantly more accurate, it is also computationally more expensive. To speed up computations, the micro-macro method may be used. The micro-macro method switches between two models of varying levels of detail allowing for larger stable time steps. In this paper we formulate the micro-macro method for shallow water moment equations. We perform a theoretical runtime analysis of the method and present a series of results for a dam break test and a wave transport test. The micro-macro method achieves a significant speed-up while retaining a sufficient level of accuracy.

Figures (11)

• Figure 1: The vertical variable is mapped from irregular (left) to uniform domain (right), from intro_shallow_water.
• Figure 1: Investigation of the matching step for $L = 2, \ldots, 6$ variables. Resulting velocity profiles (left) and detailed view of $\zeta \in [0.5, 0.7]$ (right). Increasing the number of variables leads to a more accurate matching.
• Figure 1: Micro-macro method simulation results for the dam break test. The macro model is fixed to be $L=2$ and the micro model is varied between $M= 4,\ldots, 7$. Velocity profile $u(\zeta)$ (left) and water height $h(x)$ (right).
• Figure 2: One iteration of the micro-macro method consists of micro step, restriction step, macro step, matching step, from intro_micro_macro.
• Figure 2: Micro-macro method simulation results for the dam break test. The macro model is fixed to be $L=3$ and the micro model is varied between $M = 4,\ldots, 7$. Velocity profile $u(\zeta)$ (left) and water height $h(x)$ (right).