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.