Spline Shallow Water Moment Equations

• SSWME is a shallow-flow model reduction technique that uses locally supported spline functions to capture vertical velocity profiles and account for non-smooth variations.
• The approach employs a Galerkin projection with spline ansatz functions, generating pre-computed tensors that preserve the low-dimensional structure while refining vertical details.
• Hyperbolic regularization and tailored numerical discretization ensure accurate, robust free-surface flow simulations that mitigate Gibbs-type oscillations in steep profiles.

Searching arXiv for the main paper and closely related moment-model and numerical-discretization references. Reduced models for free-surface flows address the high dimensionality of the underlying incompressible Navier–Stokes equations, which need to fully resolve the flow in vertical direction to compute the surface height. Classical Shallow Water Equations (SWE) assume a small depth-to-length ratio and use depth-averaging, but do not provide information about the vertical velocity profile variations. Spline Shallow Water Moment Equations (SSWME) were introduced as a hierarchy of shallow-flow models in which the horizontal velocity is expanded in piecewise defined spline ansatz functions, thereby combining low dimensionality with velocity profile modeling while allowing a flexible representation of profiles with lower regularity (Scholz et al., 31 Jul 2025). The construction extends the shallow water moment framework beyond global polynomial ansatz functions and introduces a regularized hyperbolic version with analytical proof of hyperbolicity for a hierarchy of high-order models; numerical simulations show high accuracy and robustness of the new models (Scholz et al., 31 Jul 2025).

1. Position within shallow-flow model reduction

The point of departure is the observation that standard depth-averaged SWE discard vertical structure, whereas shallow water moment approaches retain a finite-dimensional representation of the vertical velocity profile. In the formulation summarized for SSWME, the governing setting assumes one horizontal coordinate $x$, a flat or fixed bottom $z=h_b$, a free-surface elevation $h_s(x,t)$, and fluid depth $h=h_s-h_b$ (Scholz et al., 31 Jul 2025).

The SSWME construction is presented as a compromise between fully resolved free-surface flow models and standard reduced models. A recently proposed moment approach using Legendre polynomials as ansatz functions for vertical velocity variations had already shown the derivation of Shallow Water Moment Equations (SWME), but only global polynomials were considered so far (Scholz et al., 31 Jul 2025). SSWME replaces this global basis by spline functions with local support. This local support opens up the possibility of adaptability and greater flexibility regarding some typical profile shapes (Scholz et al., 31 Jul 2025).

This suggests that the conceptual novelty of SSWME lies less in altering the moment-closure paradigm itself than in altering the approximation space for the vertical profile. A plausible implication is that the model class is intended to improve representation of non-smooth or piecewise-structured profiles without abandoning the shallow-flow moment architecture.

2. Vertical coordinate mapping and spline ansatz

The physical vertical coordinate is mapped from

$$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$

The horizontal velocity is then approximated by

$$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$

where $u_m=\int_0^1 u\,d\zeta$ is the depth-average and the basis functions $\phi_i$ are constrained splines of zero mean,

$\int_0^1 \phi_i(\zeta)\,d\zeta=0$

(Scholz et al., 31 Jul 2025).

The spline construction begins from a uniform knot-vector

$0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$

with piecewise polynomials of degree $z=h_b$0 on each interval $z=h_b$1. The standard B-spline basis of degree $z=h_b$2, denoted $z=h_b$3, has local support on at most $z=h_b$4 adjacent intervals (Scholz et al., 31 Jul 2025). For $z=h_b$5 on $z=h_b$6 cells, one obtains $z=h_b$7 hat-functions, each supported on two adjacent cells; for $z=h_b$8 on $z=h_b$9 cells, one obtains $h_s(x,t)$0 piecewise-quadratic caps, each with support on three adjacent cells (Scholz et al., 31 Jul 2025).

To enforce the zero-mean constraint, the basis functions are formed as

$h_s(x,t)$1

so that $h_s(x,t)$2 (Scholz et al., 31 Jul 2025). The resulting $h_s(x,t)$3 inherit the local support of the B-splines, and each $h_s(x,t)$4 is nonzero on at most $h_s(x,t)$5 cells (Scholz et al., 31 Jul 2025).

The paper gives explicit examples on a uniform subdivision into two cells, $h_s(x,t)$6. For $h_s(x,t)$7 linear splines (L2, $h_s(x,t)$8, $h_s(x,t)$9),

$h=h_s-h_b$0

and for $h=h_s-h_b$1 quadratic splines (Q2, $h=h_s-h_b$2, $h=h_s-h_b$3),

$h=h_s-h_b$4

(Scholz et al., 31 Jul 2025).

These examples illustrate the central difference between spline and global-polynomial expansions: the basis is constructed from locally supported building blocks while preserving the zero-mean constraint required by the decomposition into depth-average plus moments.

3. Galerkin projection and quasi-linear system

The derivation starts from the incompressible Euler equations with hydrostatic pressure,

$h=h_s-h_b$5

After mapping $h=h_s-h_b$6 and inserting the ansatz $h=h_s-h_b$7, mass conservation becomes

$h=h_s-h_b$8

(Scholz et al., 31 Jul 2025).

The horizontal-momentum equation is then projected onto the test functions $h=h_s-h_b$9. The derivation introduces the tensors

$$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$0

together with boundary terms from a Navier slip-law at $$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$1 (Scholz et al., 31 Jul 2025).

The resulting system is an $$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$2 system in quasi-linear form for

$$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$3

$$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$4

where $$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$5, the flux-Jacobian $$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$6 has entries built from $$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$7, $$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$8, and the pre-computed tensors $$z\in[h_b,h_s(x,t)] \quad\longrightarrow\quad \zeta=\frac{z-h_b}{h(x,t)}\in[0,1].$$9, and $$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$0 collects bottom topography and viscous/friction terms. Equivalently,

$$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$1

(Scholz et al., 31 Jul 2025).

In this formulation, the spline basis affects the reduced model only through the pre-computed coefficient tensors and boundary contributions. This suggests that the SSWME hierarchy preserves the standard moment-system architecture while changing the approximation subspace and the associated algebraic couplings.

4. Hierarchy of SSWME models

Each choice of spline degree $$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$2 and number of knot intervals $$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$3 gives

$$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$4

basis functions and hence an $$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$5-th-order moment model SSWME–$$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$6 (Scholz et al., 31 Jul 2025). The hierarchy is therefore indexed by moment order but internally parameterized by the spline construction.

The paper lists several representative instances. SSWME–1 (L1) uses one linear spline,

$$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$7

In conservative-flux form,

$$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$8

(Scholz et al., 31 Jul 2025).

For SSWME–2, both two linear splines (L2) and two quadratic splines (Q2) yield $$u(x,z,t)=u(x,\zeta h+h_b,t)\approx u_m(x,t)+\sum_{i=1}^N s_i(x,t)\,\phi_i(\zeta),$$9. For L2, the last two components of the flux include

$u_m=\int_0^1 u\,d\zeta$0

plus a small non-conservative matrix $u_m=\int_0^1 u\,d\zeta$1 and friction $u_m=\int_0^1 u\,d\zeta$2 (Scholz et al., 31 Jul 2025). SSWME–3 can be constructed analogously as L3 or Q3 with $u_m=\int_0^1 u\,d\zeta$3, and all coefficients and coupling tensors are given in closed form in the appendix of the paper (Scholz et al., 31 Jul 2025).

The asymptotic statement in the paper is that, as $u_m=\int_0^1 u\,d\zeta$4 with bounded degree $u_m=\int_0^1 u\,d\zeta$5, one uniformly resolves arbitrary profiles in $u_m=\int_0^1 u\,d\zeta$6 (Scholz et al., 31 Jul 2025). Within the stated framework, this places the spline hierarchy in direct contrast with global polynomial hierarchies: the approximation space can be refined while retaining bounded local polynomial degree.

5. Hyperbolic regularization and HSSWME

Although each SSWME–$u_m=\int_0^1 u\,d\zeta$7 is formally a hyperbolic-type balance law, the matrix $u_m=\int_0^1 u\,d\zeta$8 can lose strict hyperbolicity for large moment amplitudes $u_m=\int_0^1 u\,d\zeta$9, meaning that complex eigenvalues may appear (Scholz et al., 31 Jul 2025). The paper addresses this by adopting a hyperbolic regularization following Koellermeier and Rominger, described as freezing all but the first moment to zero in the linearization of the flux-Jacobian (Scholz et al., 31 Jul 2025).

Equivalently, the moment vector $\phi_i$0 is projected onto the subspace $\phi_i$1 corresponding to a purely linear profile, that is, one Legendre mode $\phi_i$2 only. In practice this is a diagonal or similarity restriction $\phi_i$3 (Scholz et al., 31 Jul 2025). Evaluating the system matrix at $\phi_i$4 produces a family of $\phi_i$5 matrices whose characteristic polynomial factorizes in the form

$\phi_i$6

with closed-form expressions (Scholz et al., 31 Jul 2025).

For SSWME–2 (L2), the unregularized characteristic polynomial in scaled form $\phi_i$7 is reported as

$\phi_i$8

which admits complex roots unless $\phi_i$9 lie in a restricted region (Scholz et al., 31 Jul 2025). Imposing

$\int_0^1 \phi_i(\zeta)\,d\zeta=0$0

yields the hyperbolic HSSWME–2 system matrix

$\int_0^1 \phi_i(\zeta)\,d\zeta=0$1

whose eigenvalues are

$\int_0^1 \phi_i(\zeta)\,d\zeta=0$2

all real (Scholz et al., 31 Jul 2025).

A similar construction yields HSSWME–$\int_0^1 \phi_i(\zeta)\,d\zeta=0$3 for any $\int_0^1 \phi_i(\zeta)\,d\zeta=0$4, and analytically this is a similarity transformation of the hyperbolic Legendre-moment model (Scholz et al., 31 Jul 2025). This suggests that the regularization is not an ad hoc stabilization of a single low-order case, but a structural transfer of hyperbolicity from the corresponding Legendre-based formulation to the spline-based hierarchy.

6. Numerical discretization, accuracy, and robustness

The numerical discretization reported for the sample computations uses a second-order finite-volume method with HLL-type numerical fluxes, a uniform grid of approximately $\int_0^1 \phi_i(\zeta)\,d\zeta=0$5 cells, periodic boundary conditions in $\int_0^1 \phi_i(\zeta)\,d\zeta=0$6, and strong-stability-preserving Runge–Kutta time integration with CFL $\int_0^1 \phi_i(\zeta)\,d\zeta=0$7 (Scholz et al., 31 Jul 2025).

In a convergence study for a smooth travelling wave with $\int_0^1 \phi_i(\zeta)\,d\zeta=0$8, $\int_0^1 \phi_i(\zeta)\,d\zeta=0$9, $0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$0, and $0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$1, the relative $0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$2 errors in $0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$3 decay as $0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$4 increases for both linear-spline and quadratic-spline hierarchies (Scholz et al., 31 Jul 2025). The reported qualitative comparison is that quadratic-spline models $0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$5 generally outperform linear-spline models $0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$6 at the same $0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$7 (Scholz et al., 31 Jul 2025).

A separate fast-wave test uses a steep, piecewise-defined initial profile $0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$8. In that setting, Legendre expansions exhibit Gibbs-type oscillations, whereas SSWME–Q4 recovers the exact piecewise-quadratic shape (Scholz et al., 31 Jul 2025). The resulting wave height and mean-velocity propagation compare more favourably to reference low-level solutions (Scholz et al., 31 Jul 2025).

For the regularized hierarchy, the stated robustness result is that no blow-up was observed in the hyperbolically regularized models HSSWME–$0=\zeta_0<\zeta_1<\cdots<\zeta_M=1,$9, even in steep-profile runs (Scholz et al., 31 Jul 2025). Within the scope of the reported tests, this positions hyperbolic regularization as a practical complement to the spline basis: the spline construction improves profile representation, while the regularization controls loss of strict hyperbolicity.

7. Relation to Legendre-moment models and interpretive issues

SSWME is explicitly framed as an extension of the shallow water moment approach in which Legendre polynomials were previously used as ansatz functions for vertical velocity variations (Scholz et al., 31 Jul 2025). The crucial distinction is that Legendre modes are global polynomials, whereas the spline basis is piecewise defined and locally supported (Scholz et al., 31 Jul 2025). In the numerical examples, this difference is tied directly to the observation that Legendre expansions exhibit Gibbs-type oscillations for a steep, piecewise-defined profile, while a quadratic-spline hierarchy can recover the exact piecewise-quadratic shape (Scholz et al., 31 Jul 2025).

A common misconception would be to treat SSWME merely as a numerical discretization change. The derivation instead defines a different reduced-model ansatz space at the continuum level: the basis functions enter the Galerkin projection, the moment tensors, the flux structure, and the regularized system matrix (Scholz et al., 31 Jul 2025). The finite-volume scheme is therefore downstream of the model definition rather than constitutive of it.

Another possible misconception is that the spline basis by itself guarantees hyperbolicity. The paper states the opposite: even though each SSWME–$z=h_b$00 is formally a hyperbolic-type balance law, strict hyperbolicity can be lost for large moment amplitudes, and a separate hyperbolic regularization is introduced to address this (Scholz et al., 31 Jul 2025). The regularized models HSSWME–$z=h_b$01 should therefore be understood as a distinct subclass within the spline-moment framework.

More broadly, the paper’s statement that, as $z=h_b$02 with bounded degree $z=h_b$03, one uniformly resolves arbitrary profiles in $z=h_b$04 suggests a convergence-oriented interpretation of the hierarchy (Scholz et al., 31 Jul 2025). A plausible implication is that SSWME provides a route to vertically enriched shallow-flow modeling in which approximation order can be increased without resorting to globally supported polynomial bases, while retaining compatibility with moment-system regularization strategies already developed for Legendre-based models.