Shallow Water Linearized Moment Equations

• Shallow Water Linearized Moment Equations (SWLME) are hyperbolic extensions of the standard shallow water equations that resolve vertical velocity profiles via orthogonal polynomial expansion.
• The model neglects nonlinear inter-moment couplings while retaining quadratic contributions in momentum flux to derive explicit steady states and a conservative energy law.
• SWLME underpin robust numerical methods, including well-balanced finite-volume and DG schemes, to accurately simulate shallow flows with resolved vertical structure.

Searching arXiv for the cited SWLME papers to ground the article and confirm related work. Searching (Koellermeier, 2 Feb 2026, Koellermeier et al., 2020, Huang et al., 2020, Caballero-Cárdenas et al., 14 Jan 2025, Fan et al., 17 Jul 2025, Careaga et al., 6 Feb 2026). Shallow Water Linearized Moment Equations (SWLME) are a hyperbolic extension of the classical shallow water equations (SWE) designed to represent vertically varying horizontal velocity profiles within a shallow-flow framework. Instead of assuming that horizontal velocity is uniform over depth, the model expands the vertical profile in orthogonal polynomials and evolves the associated moment coefficients together with water depth and mean momentum. In the SWLME variant, nonlinear inter-moment couplings are suppressed while nonlinear mean-flow transport and quadratic moment contributions to the momentum flux are retained. This yields a model that is analytically tractable, admits explicit steady-state characterizations, and supports conservative energy identities and stable high-order discretizations (Koellermeier, 2 Feb 2026).

1. Origin and conceptual role within shallow-water moment modeling

Classical one-dimensional SWE assume that the horizontal velocity is vertically uniform and replace the true profile by its depth average. In the notation used in the recent energy analysis, the resulting system is

$$\partial_t h+\partial_x(hu_m)=0,\qquad \partial_t(hu_m)+\partial_x\!\left(hu_m^2+\frac12gh^2\right)=-gh\,\partial_x b.$$

This approximation is robust, but it can be inaccurate when the actual vertical velocity profile varies strongly with depth, which the cited literature identifies as a major source of error in open-channel flow calculations (Koellermeier, 2 Feb 2026).

Shallow Water Moment Equations (SWME) address this limitation by expanding the horizontal velocity in orthogonal basis functions over a mapped vertical coordinate and deriving additional evolution equations for the expansion coefficients. In this framework, the moment variables encode departures from the depth average and allow the model to resolve vertical shear without introducing a multilayer discretization (Koellermeier et al., 2020).

The SWLME arise as a particular SWME specialization. In the small-moment formulation, the higher-order equations are linearized under the assumption $\alpha_i=\mathcal{O}(\epsilon)$, so quadratic couplings of the form $\alpha_j\alpha_k$ and $\alpha_j\partial_x(h\alpha_k)$ are neglected in the moment subsystem while the quadratic moment contribution in the mean-momentum flux is retained (Koellermeier et al., 2020). In the notation of the energy paper, the same specialization is expressed by setting the triple-product and mixed-derivative coefficients to zero,

$A_{ijk}=0,\qquad B_{ijk}=0,$

which eliminates nonlinear self-interactions among the higher moments while preserving their linear coupling to the depth-averaged velocity $u_m$ (Koellermeier, 2 Feb 2026).

A recurrent misconception is to read “linearized” as meaning that the whole PDE system is linear. The cited papers do not use the term in that sense. “Linearized” refers specifically to the moment closure: nonlinear couplings among higher-order moments are removed, but the model still contains nonlinear advection of the mean flow and quadratic moment terms in the momentum flux (Koellermeier, 2 Feb 2026).

2. Variables, vertical reconstruction, and governing equations

The SWLME are formulated for one-dimensional shallow free-surface flow with hydrostatic pressure, shallow-water scaling, smooth time-independent bathymetry $b(x)$, and, in the energy derivation, frictionless flow. The primary variables are the water depth $h(x,t)$, the depth-averaged horizontal velocity $u_m(x,t)$, gravitational acceleration $g$, and moment coefficients $\alpha_i=\mathcal{O}(\epsilon)$0 or $\alpha_i=\mathcal{O}(\epsilon)$1, depending on notation (Koellermeier, 2 Feb 2026).

The vertical coordinate is mapped to

$\alpha_i=\mathcal{O}(\epsilon)$2

and the horizontal velocity profile is reconstructed as

$\alpha_i=\mathcal{O}(\epsilon)$3

where $\alpha_i=\mathcal{O}(\epsilon)$4 are shifted Legendre polynomials on $\alpha_i=\mathcal{O}(\epsilon)$5. Their orthogonality introduces the recurrent factors $\alpha_i=\mathcal{O}(\epsilon)$6 in fluxes, kinetic energies, and closure coefficients (Koellermeier, 2 Feb 2026). In the alternative notation used in the steady-state and stability papers,

$\alpha_i=\mathcal{O}(\epsilon)$7

The first few basis functions are explicitly given in the entropy analysis as

$\alpha_i=\mathcal{O}(\epsilon)$8

(Careaga et al., 6 Feb 2026).

In the frictionless one-dimensional form with bathymetry, the SWLME read

$\alpha_i=\mathcal{O}(\epsilon)$9

$\alpha_j\alpha_k$0

$\alpha_j\alpha_k$1

(Koellermeier, 2 Feb 2026). In the balance-law notation used for topography,

$\alpha_j\alpha_k$2

with

$\alpha_j\alpha_k$3

(Koellermeier et al., 2020).

Relative to SWE, the model changes the dynamics in two structurally important ways. First, the momentum flux gains

$\alpha_j\alpha_k$4

which represents the kinetic contribution of vertical deviations from the mean and can be interpreted as a dynamically resolved analogue of a Boussinesq shape factor. Second, the $\alpha_j\alpha_k$5 additional transport equations evolve the vertical profile itself rather than prescribing it algebraically (Koellermeier, 2 Feb 2026). The same expansion yields a profile-dependent Boussinesq coefficient

$\alpha_j\alpha_k$6

so the deviation from $\alpha_j\alpha_k$7 is controlled directly by the resolved moment amplitudes (Koellermeier, 2 Feb 2026).

3. Conservative energy law, skew-symmetric form, and entropy variables

A central recent development is the systematic derivation of the SWLME energy equation by extending the standard SWE energy argument to the moment-augmented system (Koellermeier, 2 Feb 2026). The construction proceeds by combining three balances: potential energy from continuity multiplied by $\alpha_j\alpha_k$8, mean kinetic energy from a skew-symmetrized form of the momentum equation, and partial kinetic energies from skew-symmetrized moment equations.

For each moment equation, the derivation rewrites

$\alpha_j\alpha_k$9

in skew-symmetric form, multiplies by $\alpha_j\partial_x(h\alpha_k)$0, and uses Legendre normalization to obtain the partial kinetic-energy balance

$\alpha_j\partial_x(h\alpha_k)$1

After summation and addition of the potential-energy balance, the total energy law becomes

$\alpha_j\partial_x(h\alpha_k)$2

with energy density

$\alpha_j\partial_x(h\alpha_k)$3

and energy flux

$\alpha_j\partial_x(h\alpha_k)$4

(Koellermeier, 2 Feb 2026).

This law extends the classical SWE energy identity by adding both the moment kinetic energies and their associated flux contribution. In the frictionless hydrostatic setting treated there, the energy source satisfies $\alpha_j\partial_x(h\alpha_k)$5. The same paper notes that if friction or relaxation terms are added, then under standard modeling assumptions the corresponding source is non-positive, so the total energy becomes dissipative rather than conservative (Koellermeier, 2 Feb 2026).

The derivation relies on split or skew-symmetric formulations of advective terms. For the mean velocity, the key identity averages the conservative and advective forms of momentum; for each moment, an analogous average symmetrizes the transport operator. This structure is important beyond formal analysis because split forms are a standard route to discrete conservative or non-increasing energy when paired with suitable numerical fluxes (Koellermeier, 2 Feb 2026).

The energy can also be recovered from entropy variables. Writing the conservative variables as $\alpha_j\partial_x(h\alpha_k)$6, the entropy function is

$\alpha_j\partial_x(h\alpha_k)$7

with entropy variables

$\alpha_j\partial_x(h\alpha_k)$8

The energy identity is then obtained from

$\alpha_j\partial_x(h\alpha_k)$9

which provides a compact entropy formulation of the same conservation law (Koellermeier, 2 Feb 2026). The general SWME entropy analysis subsequently showed that the total energy is an entropy function for the full moment system and that Newtonian slip and Manning friction are entropy dissipative with respect to the corresponding entropy variables (Careaga et al., 6 Feb 2026).

4. Hyperbolicity, steady states, and equilibrium stability

For the SWLME transport matrix, the cited spectral formula is

$A_{ijk}=0,\qquad B_{ijk}=0,$0

Thus the model has two gravity-wave speeds modified by the moment energy and $A_{ijk}=0,\qquad B_{ijk}=0,$1 additional convective speeds equal to the mean velocity (Huang et al., 2020). A careful reading of the equilibrium-stability analysis shows that the system is hyperbolic, with $A_{ijk}=0,\qquad B_{ijk}=0,$2 real eigenvalues and a complete set of eigenvectors, but not strictly hyperbolic because the eigenvalue $A_{ijk}=0,\qquad B_{ijk}=0,$3 has multiplicity $A_{ijk}=0,\qquad B_{ijk}=0,$4 (Huang et al., 2020).

With topography and no friction, smooth steady states satisfy

$A_{ijk}=0,\qquad B_{ijk}=0,$5

The lake-at-rest state is recovered as the special case

$A_{ijk}=0,\qquad B_{ijk}=0,$6

These identities generalize the classical steady SWE relations by adding a moment contribution to the Bernoulli-type invariant and by requiring each normalized moment $A_{ijk}=0,\qquad B_{ijk}=0,$7 to remain constant along the steady profile (Koellermeier et al., 2020).

In the presence of the Newtonian slip friction term analyzed in the equilibrium-stability paper, three equilibrium manifolds arise, corresponding to three asymptotic friction regimes. The water-at-rest equilibrium is

$A_{ijk}=0,\qquad B_{ijk}=0,$8

The constant-velocity equilibrium in the perfect-slip limit satisfies

$A_{ijk}=0,\qquad B_{ijk}=0,$9

so the velocity profile is constant in $u_m$0. The bottom-at-rest equilibrium in the no-slip limit is

$u_m$1

which enforces zero velocity at the bed because $u_m$2 (Huang et al., 2020).

The stability conclusions are not uniform across these manifolds. Yong’s structural stability conditions are satisfied for the water-at-rest and constant-velocity equilibria, which the paper interprets as a necessary condition for stable numerical solutions near those manifolds. By contrast, the bottom-at-rest equilibrium can admit unstable modes depending on the velocity profile (Huang et al., 2020). The same study gives explicit unstable examples and notes that these modes typically involve sign changes in the vertical velocity profile, that is, backflow near the bottom. The paper explicitly connects this to the physical scope of the shallow-flow approximation: such sign-changing profiles violate the intended shallow-flow regime with no small-scale vortices (Huang et al., 2020).

5. Numerical discretizations: well-balanced, semi-implicit, and entropy-stable schemes

The explicit characterization of steady states has made SWLME a natural testbed for well-balanced path-conservative discretizations. A finite-volume construction based on local stationary reconstructions was developed for the topography-dependent SWLME, with one-sided fluctuations built from reconstructed left and right states, Roe-type averages, and an HLL-like polynomial viscosity matrix. The scheme preserves lake-at-rest and general moment-bearing steady states exactly in the frictionless topography-driven setting (Koellermeier et al., 2020).

That well-balanced finite-volume framework is built around local stationary solutions defined by the constants

$u_m$3

and recovers the local depth from a quartic equation

$u_m$4

with

$u_m$5

Numerically, the paper reports that with $u_m$6 moments and $u_m$7, both first- and second-order well-balanced schemes preserve lake-at-rest and nontrivial steady states to machine precision, whereas non-well-balanced variants exhibit nonzero errors or order $u_m$8–$u_m$9 departures in $b(x)$0 for several stationary tests (Koellermeier et al., 2020).

For low-Froude regimes, a semi-implicit exactly fully well-balanced relaxation scheme was proposed. It introduces a Suliciu-type relaxation pressure $b(x)$1, splits the dynamics into acoustic and transport subsystems, treats the acoustic part implicitly, and uses Strang splitting with polynomial reconstruction to maintain second-order accuracy and exact steady-state preservation (Caballero-Cárdenas et al., 14 Jan 2025). The relaxation system enforces the subcharacteristic condition

$b(x)$2

and, in the implementation described there, uses instantaneous relaxation $b(x)$3 at each step (Caballero-Cárdenas et al., 14 Jan 2025). In the reported low-Froude tests, the implicit method allows CFL numbers around $b(x)$4–$b(x)$5 versus explicit CFL around $b(x)$6, with speedups $b(x)$7 and $b(x)$8 for one subcritical zero-moment case and $b(x)$9 and $h(x,t)$0 for a subcritical nonzero-moment case (Caballero-Cárdenas et al., 14 Jan 2025).

High-order discontinuous Galerkin formulations have been constructed in two complementary directions. First, path-conservative well-balanced DG methods were designed to preserve still-water and moving-water equilibria by working in equilibrium-preserving spaces and using hydrostatic reconstructions together with DLM-consistent path integrals (Fan et al., 17 Jul 2025). In the accuracy test reported there, $h(x,t)$1 DG achieves third-order convergence, while the equilibrium tests show machine-precision preservation of still water and moving-water states, with $h(x,t)$2 errors around $h(x,t)$3–$h(x,t)$4 depending on the case (Fan et al., 17 Jul 2025).

Second, entropy-stable DG spectral element methods were developed from the total-energy entropy structure of the SWME. The construction augments the bed as a stationary variable, embeds the topography source into the nonconservative term, derives entropy-conservative two-point fluxes satisfying a discrete entropy-flux compatibility condition, and then adds entropy-variable dissipation to obtain a semi-discrete entropy inequality (Careaga et al., 6 Feb 2026). In that framework, the lake-at-rest set

$h(x,t)$5

is preserved exactly, while numerical examples show high-order convergence, entropy dissipation under Newtonian slip and Manning friction, and robustness over long times (Careaga et al., 6 Feb 2026).

6. Comparison with related models, limitations, and active directions

SWLME occupy a specific point in a broader hierarchy of shallow-water moment models. Relative to the full nonlinear SWME, they sacrifice nonlinear moment–moment interactions to secure tractable eigenstructure, explicit steady states, and globally hyperbolic transport behavior. Relative to the classical SWE, they retain the correct mass and momentum structure while adding resolved vertical degrees of freedom. Relative to earlier hyperbolic regularizations such as HSWME and $h(x,t)$6-HSWME, they keep the exact momentum equation but advect all higher moments with the repeated convective speed $h(x,t)$7 (Koellermeier, 22 May 2025).

This modeling choice has consequences for accuracy. In the primitive-variable regularization study, a dam-break comparison showed that all tested models had relative errors below about $h(x,t)$8 across $h(x,t)$9, $u_m(x,t)$0, $u_m(x,t)$1, and $u_m(x,t)$2, but SWLME exhibited the largest errors overall, whereas PMHSWME delivered the smallest errors for most variables. The paper attributes this to over-linearization of moment transport in SWLME and to the importance of preserving the exact momentum equation in transient problems with vertical shear (Koellermeier, 22 May 2025). A plausible implication is that SWLME are particularly attractive when analytical structure and robust numerics dominate model-selection criteria, whereas primitive-variable regularizations may be preferable when stronger nonlinear profile interactions must be retained.

The model also has clear regime limitations. The derivations cited here assume hydrostatic pressure, shallow-water scaling, one horizontal dimension in the core formulations, and $u_m(x,t)$3 (Koellermeier, 2 Feb 2026). Several analyses emphasize that the linearized model is justified under small or moderate deviations from a vertically uniform profile, expressed as $u_m(x,t)$4 in the steady-state and low-Froude literature (Koellermeier et al., 2020). Strong bathymetric gradients, non-hydrostatic effects, and detailed friction models require extended balances and additional source terms (Koellermeier, 2 Feb 2026). Positivity preservation and wetting–drying treatment are not incorporated in all high-order DG formulations, which the DG literature identifies as an open implementation issue rather than a closed theoretical point (Fan et al., 17 Jul 2025).

Several current directions refine or generalize the SWLME perspective without abandoning the moment framework. One line develops modified source terms for non-slip regimes so that moment-enhanced shallow-water models remain effective when the original stiff slip source would drive the bed velocity too aggressively to zero; the key change is a finite effective friction coefficient that remains bounded in the non-slip limit (Zhou et al., 26 May 2025). A second line performs asymptotic analysis near viscous slip equilibrium and derives reduced shallow water moment equations with fewer active variables and reported computational cost reductions up to $u_m(x,t)$5 compared to SWME, while improving accuracy up to $u_m(x,t)$6 over SWE in the tests presented there (Daemen et al., 2 Mar 2026). A third line uses the hierarchical structure of moment models to adapt the model order in space and time; for one-dimensional adaptive simulations, two interface-coupling strategies and residual-based order indicators produce speedups up to $u_m(x,t)$7 relative to a fixed high-order model (Verbiest et al., 29 Oct 2025).

Within this broader landscape, SWLME remain the canonical linearized moment model: a vertically enriched extension of SWE with explicit wave speeds, explicit steady states, a conservative total-energy law in the frictionless hydrostatic regime, and a well-developed ecosystem of well-balanced, low-Froude, and entropy-stable numerical methods (Koellermeier, 2 Feb 2026).