Hierarchies of Free-Surface Flow Models

• Free-surface flow models are reduced-order formulations capturing fluid dynamics with deformable boundaries under gravity and capillarity.
• Hierarchical approaches use moment and spline expansions to represent nonuniform vertical velocity profiles with enhanced physical fidelity.
• Adaptive simulation techniques enable efficient computation across diverse flow regimes by dynamically adjusting model order based on local error estimators.

Free-surface flow models describe the dynamics of fluid domains with a deformable upper boundary subject to gravity, capillarity, and complex boundaries. Owing to the high dimensionality of the incompressible Navier–Stokes equations, various reduced-order models have been developed, exploiting the typically small vertical-to-horizontal aspect ratio in these systems. These models are naturally organized in systematic hierarchies, ranging from fully three-dimensional formulations through shallow water and thin-film equations to sophisticated moment- or spline-based reductions that can capture nontrivial vertical structure. The study of such model hierarchies is critical for efficient simulation, adaptive modeling, and physical fidelity in computational hydrodynamics, environmental mechanics, and geophysical flows.

1. Fundamental Model Reduction: From Navier–Stokes to Shallow Flows

The starting point is the three-dimensional incompressible Navier–Stokes (or equivalently, for granular matter, generalized rheological) equations: $$\begin{cases} \partial_t \mathbf u + \mathbf u \cdot \nabla \mathbf u = -\frac{1}{\rho} \nabla p + \mathbf g + \text{rheology terms} \ \nabla \cdot \mathbf u = 0 \end{cases}$$ subject to stress balances at the free surface and slip/no-slip at the bottom. Full resolution in all spatial directions is computationally prohibitive for large-scale or long-time flow scenarios. Classic reductions therefore invoke the shallow-layer or lubrication limit:

• Vertical length scale $H \ll$ horizontal length $L$ ($\epsilon = H/L \ll 1$).
• Hydrostatic (or lithostatic for granular) pressure approximations.
• Depth-averaging or vertical-moment expansion for flow variables.

Depth integration yields the classical Saint-Venant (Shallow Water) equations (SWE) for local depth $h$ and depth-averaged velocity $u_m$: $$\partial_t h + \partial_x(h u_m) = 0,\qquad \partial_t (h u_m) + \partial_x\left(h u_m^2 + \frac{g}{2}h^2\right) = \text{source/friction terms}$$ with modifications for viscosity, bottom topography, or nonhydrostatic pressures (Verbiest et al., 29 Oct 2025, Steldermann et al., 8 Jan 2026).

Thin-film models—valid for very small Reynolds number and dominated by viscous forces—lead to lubrication-type PDEs for the film height, where the velocity is eliminated in favor of a height-dependent mobility (Peschka et al., 2021). For complex topographies or granular flows, additional constraints and variables—such as basal pressure or three-dimensional depth-averaged velocity—are needed (Rauter et al., 2018).

2. Vertical Structure: Hierarchies by Moment and Spline Expansions

While classical shallow water and thin-film models assume a uniform (vertically constant) velocity profile, accuracy and physical realism improve by explicitly modeling the vertical structure of the horizontal velocity field via hierarchical moment expansions: $$u(x, z, t) = u_m(x, t) + \sum_{i=1}^M \alpha_i(x, t)\, \phi_i(\zeta)$$ where $\zeta = (z - h_b(x))/h(x, t)$ is the normalized vertical coordinate and $\phi_i$ are chosen basis functions—most commonly shifted Legendre polynomials (Steldermann et al., 8 Jan 2026, Verbiest et al., 29 Oct 2025), or, in more advanced approaches, B-splines of varying degree for localized refinement (Scholz et al., 31 Jul 2025). This leads to Shallow Water Moment Equations (SWME) or Spline Shallow Water Moment Equations (SSWME).

The resulting system has $M+2$ equations in 1D ($h,\, h u_m,\, h\alpha_1,\,\ldots,\,h\alpha_M$), conveying a model order $M$ hierarchy:

• $M=0$: Recovers classical SWE.
• $M\geq 1$: Systematically includes nonuniform profile effects, such as shear and higher-order velocity variations.

Splines, by their local support and higher flexibility, address issues inherent to global polynomials, such as Gibbs oscillations and lack of local adaptivity, while retaining a natural hierarchy: each additional basis function adds a new dynamical variable and equation to the system (Scholz et al., 31 Jul 2025).

3. Model Hierarchies: Levels and Governing Equations

The principal free-surface model hierarchy includes (ordered by modeling complexity):

Level	Primary Variables	Notes
Saint-Venant (SWE)	$h$, $u_m$	Depth-averaged, hydrostatic, no vertical structure
Thin-film (lubrication)	$h$	Viscous-dominated, mobility depends on $h$, can include capillarity
Shallow Water Moment Equations (SWME)	$h$, $u_m$, $\alpha_1, ..., \alpha_M$	Systematically increased vertical structure (polynomial moments)
Spline Shallow Water Moment Equations	$h$, $u_m$, $s_1, ..., s_N$	Piecewise-local velocity structure, regularization for hyperbolicity
Savage–Hutter Granular Model	$h$, 2D tangential velocity	Surface-aligned coordinates, curvilinear, includes metric curvature effects
3D Cartesian finite-area shallow flows	$h$, 3D depth-averaged velocity	Full 3D tangency constraint, curvature via constraints not metric tensors
Full free-boundary Stokes/Navier-Stokes	$\mathbf u$, $p$, $h$	No reduction, includes dynamic contact lines and free interfaces

This table highlights the major branches and their distinct treatment of vertical information, curvature/topography, and governing variable sets (Steldermann et al., 8 Jan 2026, Verbiest et al., 29 Oct 2025, Scholz et al., 31 Jul 2025, Peschka et al., 2021, Rauter et al., 2018).

4. Adaptive and Multiscale Simulation in Model Hierarchies

Regions of a free-surface flow may demand different model complexities—smooth parts admit low-order closures, while fronts, shocks, or instabilities require higher vertical resolution. Hierarchical models, particularly SWME and SSWME, support adaptive simulation: local model order $M$ is increased or decreased dynamically according to local error estimators derived from the hierarchy’s structure.

Adaptive schemes employ coupling at the interfaces of regions with differing model order:

• Padded Buffer Cell (PBC) approach: Expands lower-order states by padding with dynamically updated auxiliary variables, but is not globally conservative.
• Conservative Interface Flux (CIF): Embeds with zero padding for high-order variables to maintain conservation in primary quantities.

Model refinement and coarsening rely on the norms of high-order variables and their residuals, enabling dynamic enrichment or reduction on-the-fly (Verbiest et al., 29 Oct 2025). This naturally enables efficiency gains (e.g., up to 60% compared to uniformly high-order models) without sacrificing fidelity where vertical structure is physically relevant.

5. Curvature, Topography, and Three-Dimensional Effects

Accurate modeling of complex terrain and fully three-dimensional kinematics in free-surface models is nontrivial. In particular:

• Traditional approaches (e.g., Savage–Hutter) use surface-aligned curvilinear coordinates, introducing fictitious forces and complicated metric tensors.
• The 3D Cartesian finite-area approach (Rauter et al., 2018) retains the full three-dimensional structure of the depth-averaged velocity and enforces surface tangency via a pressure constraint, so curvature appears only implicitly, not in metric terms. The basal pressure, essential for pressure-dependent granular rheologies, is then determined in a coupled, dynamically consistent manner.
• Thin-film models classically impose gravity in the vertical only and incorporate curvature via capillary terms; higher-dimensional or non-planar extensions require careful geometric treatment (Peschka et al., 2021).

This suggests that modern hierarchy frameworks can unify the treatment of curvature and vertical structure by constraining velocity rather than relying upon explicit curvature terms or metric tensors.

6. Numerical Discretization and Automated Solution Frameworks

Hierarchical free-surface flow models are amenable to systematic discretization strategies, typically finite volume, finite element, or finite area methods, adapted to arbitrary mesh geometries and local model complexity (Steldermann et al., 8 Jan 2026, Rauter et al., 2018, Peschka et al., 2021). Examples include:

• Finite-area and finite-volume discretizations preserving divergence and projection structures, particularly for complex topographies (Rauter et al., 2018).
• High-order ALE finite element schemes (with mesh motion), maintaining energy stability and mass conservation in free-boundary thin-film models (Peschka et al., 2021).
• Symbolic code-generation frameworks such as Zoomy, which automate the derivation, analysis, and numerical solution of model hierarchies for any order $N$, permitting dynamic adaptation, code synthesis, and built-in stability analysis (Steldermann et al., 8 Jan 2026).

Such automated frameworks facilitate reproducible benchmarking, exploration of accuracy versus complexity trade-offs, and accelerate the implementation of novel hierarchical models in application-oriented settings.

7. Numerical Performance and Convergence Properties

Systematic benchmarking of hierarchical models demonstrates the following properties:

• Rapid convergence of moment and spline-based models for key flow quantities—e.g., $L_1$ errors in height and velocity drop by factors of $2$–$5\times$ for each increment in $N$, and quadratic-spline models (SSWME-Q) often outperform linear ones (SSWME-L) for a given number of modes (Scholz et al., 31 Jul 2025).
• Adaptive hierarchical methods achieve near-highest-order accuracy with runtime savings (up to $60\%$) compared to pure high-order models, with moment errors within a factor of $2$ of the maximal model, and order-of-magnitude improvements over the lowest-order models (Verbiest et al., 29 Oct 2025).
• Hyperbolic regularization is necessary at higher orders to guarantee well-posedness, and introduces minimal loss of accuracy compared to unregularized counterparts (Scholz et al., 31 Jul 2025).
• For two-dimensional and full three-dimensional domains, modern frameworks demonstrate scalability of these methods to unstructured grid environments and complex boundary/topography scenarios (Steldermann et al., 8 Jan 2026).

These findings substantiate the practical value of hierarchical free-surface flow models, which enable high accuracy in targeted flow regions while maintaining computational tractability across large domains.

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References:

• (Rauter et al., 2018)
• (Peschka et al., 2021)
• (Verbiest et al., 29 Oct 2025)
• (Scholz et al., 31 Jul 2025)
• (Steldermann et al., 8 Jan 2026)