Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes for Shallow Water Linearized Moment Equations

TL;DR

This work introduces second-order flux globalization based well-balanced path-conservative central-upwind schemes for the hyperbolic shallow water linearized moment equations (HSWLME), explicitly addressing nonconservative terms and friction. By reformulating the system in a quasi-conservative form with a global flux and reconstructing equilibrium variables, the method preserves a broad class of steady states, including moving-water equilibria. The approach combines WB equilibrium reconstruction, path-conservative integration for nonconservative terms, and a semi-discrete evolution to achieve robust, accurate solutions, demonstrated on convergence to steady states, small perturbations, and dam-break problems. The results suggest improved stability and fidelity over non-WB schemes, with potential impact on reliable simulations of shallow water flows that require higher-order vertical resolution.

Abstract

We develop second-order path-conservative central-upwind (PCCU) schemes for the hyperbolic shallow water linearized moment equations (HSWLME), which are an extension of standard depth-averaged models for free-surface flows. The proposed PCCU schemes are constructed via flux globalization strategies adapted to the nonconservative form via a path-conservative finite-volume method. The resulting scheme is well-balanced (WB) in the sense that it is capable of exactly preserving physically relevant steady states including moving-water ones. We validate the proposed scheme on several benchmarks, including smooth solutions, small perturbation of steady states, and dam-break scenarios. These results demonstrate that our flux globalization based WB PCCU schemes provide a reliable framework for computing solutions of shallow water moment models with nonlinear and nonconservative features.

Figures (8)

• Figure 4.1: Example 1, Case (a): $h$, $q$, $E$, $E_1$, and $E_2$ computed by the WB and NWB PCCU schemes over the continuous (top two rows) and discontinuous (bottom two rows) $Z$.
• Figure 4.2: Same as in Figure \ref{['fig41']}, but for Case (b).
• Figure 4.3: Same as in Figures \ref{['fig41']} and \ref{['fig42']}, but for Case (c).
• Figure 4.4: Example 2 (small perturbation): The difference $h(x,t)-h_{\rm eq}(x)$ computed using $100$ and $1000$ uniform cells at times $t=0.8$ in Case (a) and $t=2$ in Cases (b) and (c).
• Figure 4.5: Example 3: $h$, $u$, $\alpha_1$, and $\alpha_8$ computed by the WB PCCU scheme. $\nu=0$.

Theorems & Definitions (2)

• Remark 3.1
• Remark 3.2