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Two-Dimensional Shallow Water Linearized Moment Equations: Hyperbolicity and Well-Balanced Schemes

Shiping Zhou, Juntao Huang, Andrew Christlieb

arXiv:2609.28800v1math.NAcs.NA

TL;DR

The direct two-dimensional extension of shallow water linearized moment equations can have real speeds but lack a complete eigenbasis. The paper constructs a rotationally invariant modification and midpoint-collocated, well-balanced schemes; tests report equilibrium preservation and accuracy results, alongside comparisons with three-dimensional simulations.

  • Problem

    The direct two-dimensional SWLME extension can have real characteristic speeds but lack a complete eigenbasis, complicating global hyperbolicity across directions.

    Source: §3.2 · p. 9, §3.3 · p. 11

  • Method

    The paper constructs a rotationally invariant globally hyperbolic modification and path-conservative finite-volume schemes that preserve a midpoint-collocated discrete moving equilibrium.

    Source: §1 · p. 2, §6 · p. 41

  • Results

    Tests show roundoff equilibrium preservation, second-order stationary-branch approximation, and near-second-order accuracy for tested perturbations; OpenFOAM comparisons show improved rotational profiles with nonzero moments but remaining radial-motion differences.

    Source: §6 · p. 41

  • Takeaways & Limitations

    Retaining nonzero moments improves reconstructed rotational velocity profiles in the comparisons, while differences in radial motion remain.

    Source: §6 · p. 41

    Reference-grid sensitivity is non-negligible for some moment components and is reported alongside time-step sensitivity.

    Source: §5.4 · p. 32

Abstract

from arXiv · show

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.

1 Introduction

Two-dimensional shallow water linearized moment equations must address directional hyperbolicity while representing velocity moments and nonconservative terms in moving equilibria. The paper establishes a rotationally invariant hyperbolic modification and develops well-balanced schemes for a prescribed quasi-two-dimensional equilibrium.

Prior well-balanced methods

  • The introduction distinguishes prior two-dimensional lake-at-rest well-balancing from preservation of frictional moving equilibria.For one-dimensional SWLME, prior work addressed frictionless moving equilibria and later extended fully well-balanced reconstruction to smooth topography with friction.Source: §1 · p. 2

Structural contribution

  • For every retained order N ≥1 and positive depth, G-SWLME has a complete real eigenbasis, including at states where the direct extension is defective.The direct two-dimensional extension can have real characteristic speeds without a complete eigenbasis.Source: §1 · p. 2

Well-balanced schemes

  • First- and second-order path-conservative finite-volume schemes exactly preserve a midpoint-collocated discrete moving equilibrium for quasi-two-dimensional flows with Navier-slip friction and non-flat topography.The construction applies on smooth stationary branches where the stationary coefficient matrix remains invertible.Source: §1 · p. 2

Numerical comparisons

  • The numerical comparisons illustrate model differences in radial collapse.The paper also positions moment models against depth-averaged and multilayer descriptions, which present distinct trade-offs in vertical structure and coupling.Source: §1 · pp. 1, 3

2 Shallow water moment models

The section reviews the hydrostatic formulation and derives shallow water moment equations by vertically normalizing the flow and projecting its velocity profiles onto Legendre modes.

2.1 Hydrostatic equations in normalized vertical coordinates

  • The hydrostatic derivation assumes mild bottom slopes and uses stress-free surface conditions with componentwise Navier slip at a smooth stationary bottom.The slip conditions are exact for a flat bottom and approximate the tensorial Navier law to leading order for smooth mild slopes.Source: §2.1 · p. 3

2.2 Shallow water moment equations

  • Horizontal velocity profiles are approximated with Legendre expansions, whose coefficients describe vertical deviations from the depth averages.Integrating the normalized mass and momentum equations and projecting against the Legendre modes yields the shallow water moment equations.Source: §2.2 · pp. 4–5

2 Shallow water moment models

  • The two-dimensional shallow water moment model is presented in a matrix-vector formulation, with state variables comprising depth, depth-averaged velocities, and moment coefficients.The case N = 0 recovers the classical shallow water equations.Source: §2.2 · p. 5

3 Two-dimensional SWLME and hyperbolicity

This section examines the two-dimensional SWLME and its hyperbolicity, including a rotationally consistent modification of the direct formulation. The modified system is globally hyperbolic for every admissible state with positive water depth, with real-diagonalizable directional matrices even when α = 0.

3.1 Shallow water linearized moment equations

  • The two-dimensional SWLME simplifies higher-moment couplings while retaining the complete quadratic mean-momentum flux tensor, whose components form a rotationally covariant tensor.The model extends the linearized moment setting to two horizontal dimensions with Navier-slip bottom friction.Source: §3.1 · p. 7

3.2 Rotational invariance

  • The SWLME coefficient matrices satisfy rotational invariance, reducing hyperbolicity analysis to the x-direction; rotationally consistent block corrections preserve that invariance.The reduction follows because directional matrices are similar to the x-direction matrix at a rotated state.Source: §3.2 · pp. 8–9

3.3 Hyperbolicity of the unmodified SWLME

  • The direct SWLME is not globally hyperbolic: when all α moments vanish and some β moment is nonzero, eigenvalue um has algebraic multiplicity 2N + 1 but geometric multiplicity 2N.The defect occurs in the x-direction coefficient matrix.Source: §3.3 · p. 12

3.4 A globally hyperbolic modification of the SWLME

  • The first change removes the defective coupling at α = 0, and the rotationally consistent system eAG is real diagonalizable for every h > 0 and G > 0.Source: §3.4 · p. 13
  • For N ≥ 1 and G > 0, the correction blocks satisfy rotational consistency and make every directional coefficient matrix real diagonalizable, yielding global hyperbolicity of the two-dimensional G-SWLME.The result holds for every admissible state and every unit direction.Source: §3.4 · p. 14
  • At α = 0, the repeated eigenvalue u_m has a complete eigenspace, so the modified system remains pointwise hyperbolic at these states.The proof gives a 2N + 1-dimensional eigenspace for u_m when S_α = 0.Source: §3.4 · p. 14

4 Well-balanced schemes for quasi-two-dimensional flows

The section constructs first- and second-order path-conservative finite-volume schemes for quasi-two-dimensional G-SWLME flows, preserving one prescribed discrete moving equilibrium. It builds that equilibrium by midpoint collocation and uses equilibrium-deviation reconstruction with a volume correction.

4 Well-balanced schemes for quasi-two-dimensional flows

  • The methods store the equilibrium built from boundary data and reconstruct deviations from its cell averages, combining path-conservative discretization with midpoint collocation.The prescribed moving branch uses smooth stationary solutions for which the stationary coefficient matrix remains invertible.Source: §4 · p. 15, §4.1 · p. 16

4.1 Construction of a prescribed discrete equilibrium

  • The construction restricts moving equilibria to quasi-two-dimensional flows, reducing the stationary equations to an ODE, and assumes the prescribed branch avoids singular states.The branch is propagated from prescribed interface data by midpoint collocation; lake-at-rest profiles are prescribed analytically.Source: §4 · p. 16, §4.1 · p. 16

4.2 Equilibrium-deviation reconstruction and HLL fluctuations

  • Pathwise HLL speed bounds enclose every characteristic speed along the conservative straight path, and the resulting fluctuations satisfy path consistency.The method assumes positive-depth admissible states at all stages and has no positivity-preserving limiter or wetting–drying treatment.Source: §4.2 · pp. 17–18

4.3 Volume correction, time integration, and preservation

  • The equilibrium-subtracted volume correction is needed at both spatial orders, while the second-order method is formally second order for smooth solutions under a stable step with Δt = O(Δx).The consistency argument excludes limiter extrema and discontinuities and does not establish stability or positivity.Source: §4.3 · pp. 19–20
  • Both spatial orders preserve the prescribed discrete equilibrium exactly in exact arithmetic when the stored data and equilibrium-related functions and states are used consistently.With identical stored interface states and consistent equilibrium subtraction, the residual at the stored branch is limited by roundoff; this does not measure the stored branch's accuracy.Source: §4.3 · pp. 20–21

5 Numerical experiments

The numerical experiments assess two-dimensional moment dynamics and the well-balanced schemes, then compare radial-collapse results with three-dimensional OpenFOAM simulations.

5 Numerical experiments

  • The experiments compare moment-model solutions with three-dimensional OpenFOAM simulations in radial-collapse problems initialized from rest or with vertical shear.The stated test program also includes mesh refinement against a vertically resolved hydrostatic reference.Source: §5 · p. 21
  • The well-balanced tests examine preservation of lake-at-rest and prescribed quasi-two-dimensional moving equilibria, followed by finite-amplitude and weak perturbations.Source: §5 · p. 21
  • The experiments first assess two-dimensional moment dynamics through mesh refinement and comparison with a vertically resolved hydrostatic reference.Source: §5 · p. 21

5.1 Two-dimensional moment dynamics

The two-dimensional G-SWLME tests show approximately second-order spatial convergence for N = 2 and generally closer agreement with the vertically resolved reference at N = 2 than at N = 0. Higher moment orders also improve the representation of vertical velocity structure in the tested smooth flow.

5.1 Two-dimensional moment dynamics

  • Approximately second-order spatial convergence is observed for every conservative component of the N = 2 model.The N = 0 and N = 1 models exhibit similar convergence rates.Source: §5.1 · p. 22, Table 3 · p. 23
  • At t = 0.1, the N = 2 model generally agrees more closely with the reference than N = 0, although improvement varies across components and locations.Source: Figure 1 · p. 23, §5.1 · p. 24
  • Along the diagonal transect, all three orders closely match depth and depth-averaged velocities; N = 1 and N = 2 reproduce first moments, and N = 2 captures second moments.Source: §5.1 · pp. 24–25, Figure 2 · p. 24, Figure 3 · p. 24
  • The reconstructed profiles progress from depth-uniform velocities at N = 0 to linear variation at N = 1 and curvature at N = 2.At selected locations, quadratic reconstructions closely follow the reference; reference-grid sensitivity limits quantitative interpretation of the smallest componentwise differences.Source: §5.1 · p. 25, Figure 5 · p. 25

5.2 Lake-at-rest preservation

The first- and second-order well-balanced methods preserve the tested lake-at-rest equilibrium over non-flat topography, while the standard HLL method develops nonzero errors. The well-balanced methods retain zero computed free-surface and velocity errors through t = 10, including with the second-order reconstruction and time integration.

5.2 Lake-at-rest preservation

  • The test uses a prescribed rest state over non-flat topography for the quasi-two-dimensional N = 2 system, with zero friction and inverse Reynolds number.Source: §5.2 · p. 25
  • WB1 and WB2 preserve the lake-at-rest equilibrium with zero computed free-surface and velocity errors through t = 10.WB2 preserves the equilibrium to machine precision across tested meshes, including with second-order reconstruction and SSP-RK2 time integration.Source: §5.2 · pp. 27–28, Table 5 · p. 28
  • The HLL baseline does not preserve the equilibrium on any tested mesh, although its errors decrease under mesh refinement.At t = 10, its maximum absolute free-surface error is 10^-6.Source: §5.2 · pp. 27–28, Figure 7 · p. 27

5.3 Moving-equilibrium preservation

Midpoint collocation constructs frictionless and dissipative discrete moving equilibria, and WB1/WB2 preserve both to machine precision. The stored branches converge at second order, while HLL drift decreases at approximately first order.

5.3 Moving-equilibrium preservation

  • The tests use midpoint-collocation equilibria initialized from fixed boundary states and compare their stored values with independent stationary references.The frictionless case has vanishing transverse components; the dissipative case includes transverse velocity and both moment families.Source: §5.3 · p. 28, §5.3.1 · p. 29, §5.3.2 · p. 30

5.3.1 Frictionless moving equilibrium

  • The frictionless discrete equilibrium converges at second order in every nonzero component, with transverse components identically zero.The discrete and independent reference free surfaces are visually indistinguishable on the finest mesh.Source: §5.3.1 · p. 29, Figure 9 · p. 29

5.3.2 Dissipative moving equilibrium

  • The dissipative equilibrium construction converges at second order in all seven primitive components.The independent reference remains fully wet and noncritical.Source: §5.3.2 · p. 31, Figure 11 · p. 31

5.3 Moving-equilibrium preservation

  • WB1 and WB2 preserve both frictionless and dissipative stored equilibria to machine precision, while HLL drift decreases at approximately first order under refinement.The frictionless and dissipative cases cover vanishing and nonzero transverse components, respectively.Source: §5.3.2 · pp. 30, 32, §5.3.1 · p. 30

5.4 Perturbation accuracy

The tests perturb a dissipative moving equilibrium at finite and weak amplitudes to compare HLL, WB1, and WB2. WB2 gives the smallest depth errors and approaches second-order convergence in both cases, while reference-grid sensitivity affects some moment-component comparisons.

5.4 Perturbation accuracy

  • Reference-grid sensitivity is non-negligible for some moment components and is reported alongside time-step sensitivity.The perturbation errors are measured against a numerical reference, block-averaged onto the comparison mesh.Source: §5.4 · p. 32

5.4.1 Finite-amplitude perturbation

  • For the finite-amplitude perturbation, WB2 better captures the bump's amplitude and shape, while HLL and WB1 broaden it and underestimate its peak.HLL also introduces a pronounced spurious disturbance.Source: §5.4.1 · p. 33, Figure 13 · p. 33

5.4.2 Weak perturbation

  • For the weak perturbation, HLL disturbance dominates the weaker physical signal, while WB1 and WB2 resolve it and WB2 stays close to the reference.WB1 underestimates the bump's peak.Source: §5.4.2 · pp. 33–34, Figure 14 · p. 33

5.4 Perturbation accuracy

  • WB2 gives the smallest depth errors and approaches second-order convergence for both finite and weak perturbations.HLL and WB1 approach first-order convergence; WB1 has substantially smaller errors than HLL for the finite-amplitude case.Source: §5.4.1 · p. 33, Figure 14 · p. 33, Figure 13 · p. 33

5.5 Radial collapse over a perfect-slip bed

The radial-collapse tests compare reduced moment models with three-dimensional OpenFOAM, first from rest and then with initial vertical shear. The reduced models reproduce key bulk patterns and rotational profiles, but discrepancies persist in radial motion and near collapse waves.

5.5.1 Collapse from rest: the invariant SWE limit

  • For collapse from rest, OpenFOAM depth profiles and principal wave locations are nearly coincident across vertical resolutions, although local oscillation amplitudes retain some mesh sensitivity.The oscillations may include both physical undular-bore structures and numerical contributions.Source: §5.5.1 · p. 35, Figure 15 · p. 35
  • For collapse from rest, the reduced equations capture principal wave locations and large-scale depth-averaged velocity patterns, while OpenFOAM shows vertical velocity variation absent from the plug-profile reduced solutions.Initially zero moments remain zero, so the reduced-model orders coincide with the SWE solution.Source: §5.5.1 · pp. 34, 36, 37, Figure 18 · p. 37

5.5.2 Collapse with initial vertical shear

  • With initial vertical shear, water depth and depth-averaged velocities agree closely with OpenFOAM, and the N = 0, 1, 2 predictions nearly overlap.On the near-center slice, the mean velocities approximately represent radial and azimuthal motion; additional OpenFOAM oscillations concentrate in depth and radial motion.Source: Figure 19 · p. 38, §5.5.2 · p. 39
  • Agreement in α1 and α2 is close away from collapse waves, but OpenFOAM variations are larger near the waves and increasing from N = 1 to N = 2 does little to improve α1.On this slice, αj primarily describes vertical variations in radial motion; remaining differences may reflect model approximation, discretization, and water-volume projection.Source: §5.5.2 · pp. 39–40
  • The reduced models closely reproduce the amplitudes and shapes of β1 and β2, supporting their representation of the evolving rotational shear.These coefficients primarily describe vertical variations in azimuthal motion on the slice.Source: Figure 20 · p. 39, §5.5.2 · p. 40
  • In the central rotating region, higher-order reconstructions through N = 4 capture both velocity-profile shapes, with persistent ũ offset versus OpenFOAM and closer ˜v shape and magnitude agreement; N = 0 and N = 1 cannot represent observed curvature.Additional basis functions improve reconstruction, particularly near-bottom ˜v values.Source: §5.5.2 · p. 41
  • More basis functions improve complex-profile representation but do not replace an appropriate physical boundary condition; thin boundary layers may require substantially more modes.The central-profile comparison includes N = 0, 1, 2, 3, 4.Source: §5.5.2 · p. 41

6 Conclusions

The paper establishes rotational invariance and constructs a globally hyperbolic two-dimensional SWLME modification, then develops well-balanced schemes for quasi-two-dimensional flows. Tests support equilibrium preservation and accuracy within the reported settings, while OpenFOAM comparisons leave radial-motion differences and reference convergence unresolved.

6 Conclusions

  • The paper establishes rotational invariance, identifies loss of global hyperbolicity in the direct two-dimensional SWLME, and constructs the globally hyperbolic G-SWLME modification.For quasi-two-dimensional flows, first- and second-order path-conservative schemes exactly preserve a prescribed discrete moving equilibrium with Navier-slip friction and non-flat topography.Source: §6 · p. 41
  • Tests confirm roundoff-level equilibrium preservation, second-order stationary-branch approximation, and near-second-order accuracy for tested perturbations with the second-order scheme.OpenFOAM comparisons illustrate the invariant SWE limit and improved rotational velocity profiles with moments, while radial-motion differences remain and full reference convergence is unverified.Source: §6 · p. 41

Data and code availability

The data-generation source code, numerical solvers, and OpenFOAM case configurations are publicly available.

Data and code availability

  • The public repository includes data-generation code, numerical solvers, OpenFOAM case configurations, dependencies, and local simulation instructions; it is archived on Zenodo as release v1.0 [41].https://github.com/zhou-shiping/swe-well-balanced-2026Source: Data and code availability · p. 41

A Proof of rotational invariance

The appendix proves rotational invariance for the SWLME conservative fluxes and non-conservative matrices. It does so by rotating the velocity and moment pairs and verifying their corresponding identities.

A Proof of rotational invariance

  • The proof rotates the velocity and each moment pair by the same angle, then combines the conservative and non-conservative identities to establish Theorem 3.1.The rotated state is T(θ)U, with u, v, and each α_j, β_j pair transformed together.Source: A Proof of rotational invariance · pp. 41–43
  • The conservative fluxes FL(U) and GL(U) satisfy the rotational-invariance identity for every angle and admissible state.The proof verifies the identity component by component and then assembles the identities for all moment pairs.Source: A Proof of rotational invariance · pp. 42–43
  • The non-conservative matrices P1(U) and Q1(U) satisfy a corresponding rotational-invariance identity for every angle and admissible state.Their repeated 2 × 2 blocks allow the proof to verify the relation for one block pair and extend it across all moment pairs.Source: A Proof of rotational invariance · p. 43
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