We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic the continuous entropy analysis on the semi-discrete level with the DGSEM constructed on Legendre-Gauss-Lobatto (LGL) nodes. The use of LGL nodes endows the collocated nodal DGSEM with the summation-by-parts property that is key in the discrete analysis. The approximation exploits an equivalent flux differencing formulation for the volume contributions, which generate an entropy conservative split-form of the governing equations. A specific combination of an entropy conservative numerical surface flux and discretization of the nonconservative terms is then applied to obtain a high-order path-conservative scheme that is entropy conservative and has the well-balanced property for discontinuous bathymetry. Dissipation is added at the interfaces to create an entropy stable approximation that satisfies the second law of thermodynamics in the discrete case. We conclude with verification of the theoretical findings through numerical tests and demonstrate results about convergence, entropy stability and well-balancedness of the scheme.

翻译：我们提出了一种适用于二维曲线网格上双层浅水方程的熵稳定节点间断伽辽金谱元法。通过在勒让德-高斯-洛巴托节点上构建DGSEM，在半离散层面模拟连续熵分析。LGL节点的使用赋予共位节点DGSEM求和-分块性质，这对于离散分析至关重要。该近似方法利用体积贡献的等价通量差分形式，生成了控制方程的一个熵守恒分裂形式。通过将熵守恒数值表面通量与非守恒项离散化进行特定组合，我们得到了一种高阶路径守恒格式，该格式兼具熵守恒性质，并对间断地形具有完全平衡特性。在界面处引入耗散项以生成离散情形下满足热力学第二定律的熵稳定近似。最后通过数值测试验证理论发现，并展示了该格式在收敛性、熵稳定性及完全平衡性方面的结果。