Entropy analysis and entropy stable DG methods for the shallow water moment equations

We demonstrate that the shallow water moment equations satisfy an auxiliary entropy conservation law, where the entropy function corresponds to the total energy. Additionally, we show that the classical Newtonian slip friction and Manning friction terms are entropy dissipative with respect to the developed entropy variables. The results from the continuous entropy analysis are used to construct an entropy stable and well-balanced nodal discontinuous Galerkin spectral element method for the spatial approximation. Key to ensure the entropy stability of the scheme is the derivation of entropy conservative numerical fluxes that satisfy a discrete version of the entropy flux compatibility condition. Finally, numerical examples demonstrate the performance of the scheme and validate the theoretical results.

💡 Research Summary

This paper investigates the entropy structure of the one‑dimensional shallow water moment equations (SWME) and leverages that structure to develop an entropy‑stable discontinuous Galerkin (DG) spectral element method. The SWME extend the classical shallow water equations (SWE) by representing the vertical velocity profile as a polynomial expansion; the coefficients of this expansion are the moments α₁,…,α_N together with the water depth h and the depth‑averaged velocity u_m. The governing equations (3a‑3c) consist of a continuity equation, a momentum equation, and N moment equations that contain nonlinear coupling terms A_i and B_i, the latter being non‑conservative.

The authors first perform a continuous entropy analysis. By deriving averaged kinetic‑energy and potential‑energy equations (Lemma 1) and an auxiliary equation for the term Ψ that depends on the moments (Lemma 2), they obtain a total kinetic‑energy balance that still contains a non‑conservative contribution Q. Lemma 3 shows that Q can be written as the spatial derivative of a scalar function b_Q, thereby converting the whole kinetic‑energy balance into a conservative form. Adding the potential‑energy equation yields a total‑energy conservation law (Theorem 1) of the form ∂ₜE(u)+∂ₓF(u)=0, where

E(u)=½ h u_m² + ∑_{i=1}^N α_i²/(2i+1) + ½ g h² + g h b

and

F(u)=½ h u_m³ + ½ u_m ∑{i=1}^N α_i²/(2i+1) + g h u_m (h+b) + ½ ∑{i,j,k} (A_{ijk}+B_{ijk}) h α_i α_j α_k.

The total energy E is convex with respect to the conservative variables u, making it a mathematical entropy. The associated entropy variables w are obtained by differentiating E with respect to u, resulting in w₁=−½ u_m²−∑α_i²/(2i+1)+g(h+b), w₂=u_m, and w_{i+2}=α_i/(2i+1).

Next, the paper examines two widely used friction models. The Newtonian slip friction (24) combines a bottom‑slip term proportional to (u_m+1ᵀα) and a bulk viscous term involving the matrix C̃. Lemma 4 proves that wᵀS_Ns ≤ 0, i.e., the slip friction is entropy‑dissipative. The Manning friction (28) introduces a depth‑dependent resistance term proportional to h^{-1/3}. Lemma 5 shows that this term is also entropy‑dissipative. Consequently, both friction models act as source terms that decrease the entropy, consistent with physical energy loss.

For the spatial discretisation, the authors adopt a nodal DG spectral element framework. They construct entropy‑conservative numerical fluxes that satisfy Tadmor’s entropy compatibility condition, ensuring that the semi‑discrete scheme preserves the discrete entropy balance in the absence of dissipation. To achieve entropy stability, they augment the interface fluxes with a suitable amount of numerical dissipation (e.g., a local Lax‑Friedrichs term) that guarantees wᵀ S ≤ 0 at element boundaries. The scheme is also made well‑balanced: the discretisation of the bathymetry term and the source terms is designed so that steady‑state solutions (still water over variable bottom) are exactly preserved.

A series of numerical experiments validates the theoretical developments. Test cases include: (i) a still‑water equilibrium to verify the well‑balanced property; (ii) a dam‑break problem with sharp gradients to assess shock‑capturing and entropy decay; (iii) flows with Newtonian slip friction and Manning friction to demonstrate entropy dissipation in practice; and (iv) higher‑order moment models (N=2) to illustrate the method’s capability to handle additional degrees of freedom. In all cases the discrete entropy decreases monotonically, and the DG solution exhibits reduced spurious oscillations compared with standard (non‑entropy‑stable) DG schemes. The results confirm that the proposed entropy‑stable DG method faithfully reproduces the continuous entropy behavior of the SWME, even in the presence of complex non‑conservative products and frictional source terms.

In conclusion, the paper provides a rigorous continuous entropy analysis for the SWME, proves that common friction terms are entropy‑dissipative, and translates these insights into a practical, high‑order, entropy‑stable DG spectral element method that is both well‑balanced and capable of handling the nonlinear coupling inherent in moment‑based shallow water models. Future work may extend the approach to two‑dimensional domains, incorporate more general bottom topographies, and explore adaptive mesh refinement within the entropy‑stable framework.