Two-Dimensional Shallow Water Linearized Moment Equations: Hyperbolicity and Well-Balanced Schemes
Abstract
We study rotational invariance, hyperbolicity, and well-balanced discretization of shallow water linearized moment equations in two horizontal dimensions. Although the direct extension has real characteristic speeds, its directional matrices can lack a complete eigenbasis. We construct a rotationally invariant modification and prove its global hyperbolicity for all states with positive water depth. For quasi-two-dimensional flows over non-flat topography with Navier-slip friction, we develop first- and second-order path-conservative finite-volume schemes that exactly preserve a discrete moving equilibrium constructed by midpoint collocation. Numerical tests confirm equilibrium preservation to roundoff, second-order approximation of smooth stationary branches, and near-second-order accuracy of the second-order scheme for the tested perturbations. Comparisons with three-dimensional two-phase OpenFOAM simulations illustrate the invariant shallow-water limit and show that retaining moments improves the reconstruction of curved rotational velocity profiles, while differences remain in radial motion.
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BibTeX
@misc{zhou2026twodimensional,
title = {{Two-Dimensional Shallow Water Linearized Moment Equations: Hyperbolicity and Well-Balanced Schemes}},
author = {Shiping Zhou and Juntao Huang and Andrew Christlieb},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28800},
eprint = {2609.28800},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28800},
}
RIS
TY - UNPB
TI - Two-Dimensional Shallow Water Linearized Moment Equations: Hyperbolicity and Well-Balanced Schemes
AU - Shiping Zhou
AU - Juntao Huang
AU - Andrew Christlieb
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28800
UR - https://arxiv.org/abs/2609.28800
AB - We study rotational invariance, hyperbolicity, and well-balanced discretization of shallow water linearized moment equations in two horizontal dimensions. Although the direct extension has real characteristic speeds, its directional matrices can lack a complete eigenbasis. We construct a rotationally invariant modification and prove its global hyperbolicity for all states with positive water depth. For quasi-two-dimensional flows over non-flat topography with Navier-slip friction, we develop first- and second-order path-conservative finite-volume schemes that exactly preserve a discrete moving equilibrium constructed by midpoint collocation. Numerical tests confirm equilibrium preservation to roundoff, second-order approximation of smooth stationary branches, and near-second-order accuracy of the second-order scheme for the tested perturbations. Comparisons with three-dimensional two-phase OpenFOAM simulations illustrate the invariant shallow-water limit and show that retaining moments improves the reconstruction of curved rotational velocity profiles, while differences remain in radial motion.
ER -