Entropy stable hydrostatic reconstruction schemes for shallow water systems

Abstract

In this work, we develop a new hydrostatic reconstruction procedure to construct well-balanced schemes for one and multilayer shallow water flows, including wetting and drying. Initially, we derive the method for a path-conservative finite volume scheme and combine it with entropy conservative fluxes and suitable numerical dissipation to preserve an entropy inequality in the semi-discrete case. We then combine the novel hydrostatic reconstruction with a collocated nodal split-form discontinuous Galerkin spectral element method, extending the method to high-order and curvilinear meshes. The high-order method incorporates an additional positivity-limiter and is blended with a compatible subcell finite volume method to maintain well-balancedness at wet/dry fronts. We prove entropy stability, well-balancedness, and positivity-preservation for both methods. Numerical results for the high-order method validate the theoretical findings and demonstrate the robustness of the scheme.

Figures (10)

• Figure 1: Illustration of the multilayer shallow water system with four layer and bottom topography.
• Figure 2: Illustration of bottom topography and layer heights before (a) and after the reconstruction procedure (b) at a wet/dry transition.
• Figure 3: Illustration of the bottom topography reconstruction for the different configurations (a) wet, (b) dry and (c) partially dry
• Figure 4: Spectral convergence results in space for the $L_2$-Error in quantities $h_1$, $h_2$ and $h_3$ over polynomial degree $N$ and $\Delta t=10^{-4}$
• Figure 5: Initial condition and temporal evolution of the free surface for a two-layer system with perturbed lake-at-rest setup with $N=1$, $\text{CFL}=0.7$ on $100$ equidistant elements.

Theorems & Definitions (27)

• Lemma 1
• proof
• Lemma 2: Well-balancedness
• proof
• Lemma 3: Well-balancedness ML-SWE
• proof
• remark 1
• Lemma 4: Entropy conservation (SWE)
• proof
• Lemma 5: Entropy conservation (ML-SWE)