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

TL;DR

The paper establishes a coherent entropy framework for the shallow water moment equations (SWME) by deriving the total energy-based entropy $\mathbb{E}$ and the associated entropy variables $\boldsymbol{w}$, and proves that the common friction terms dissipate entropy. Leveraging this continuous structure, it designs an entropy-conservative discontinuous Galerkin spectral-element method (DGSEM) for SWME and extends it to an entropy-stable version by adding dissipation in entropy variables, all while preserving a well-balanced lake-at-rest state. The hierarchical SWME decomposition into SWE and SWLME components enables the construction of EC fluxes and a semi-discrete entropy inequality, with numerical experiments validating entropy dissipation, accuracy, well-balancing, and robustness to topography. This work provides a high-order, entropy-stable numerical framework for SWME with nonconservative terms and bottom topography, with potential extensions to higher dimensions, non-hydrostatic regimes, and sediment transport coupling.

Abstract

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.

Figures (5)

• Figure 1: Example 1: Numerical solutions (in primitive variables) of the SWME \ref{['sys:SWME']} at $t=2$ with $N = 2$ and source term given by \ref{['eq:friction:Nslip']} with $\lambda = 0.1$ and $\nu = 0.1$, for different polynomial degrees $P$. In all simulations $K=256$ and $\text{CFL = 0.9}$.
• Figure 2: Example 1: Numerical solutions (in primitive variables) of the SWME \ref{['sys:SWME']} at $t=2$ with $N = 2$ and source term given by \ref{['eq:friction:Nslip']} with $\lambda = 0.1$ and $\nu = 0.1$, for different numbers of elements $K$. In all simulations, the polynomial degree is $P = 2$ and $\text{CFL = 0.9}$.
• Figure 3: Example 2: Entropy dissipation for the Newtonian slip friction $\mathbf{S}_{\rm Ns}$\ref{['eq:friction:Nslip']} and Newtonian Manning friction $\mathbf{S}_{\rm NM}$\ref{['eq:friction:NManning']} with respect to the entropy variables $\boldsymbol{w}$ defined in Corollary \ref{['cor:entropy:variables']}. In all simulations $K=256$ and $\text{CFL = 0.9}$.
• Figure 4: Example 4: Numerical solutions of the SWME \ref{['sys:SWME']} for the lake-at-rest test case at $t=8000$ using the well-balanced (WB) and the non well-balanced (NWB) scheme. Results are obtained on $K=64$ elements with polynomial degree $P=1$ and $\text{CFL}=0.9$.
• Figure 5: Example 4: Time series data for the total entropy and the lake-at-rest error for the lake-at-rest test case using the well-balanced (WB) and the non well-balanced (NWB) scheme. Results are obtained on $K=64$ elements with polynomial degree $P=1$ and $\text{CFL}=0.9$.

Theorems & Definitions (21)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof
• Corollary 1
• proof