标题： 一种用于曲线网格上双层浅水方程的熵稳定不连续伽辽金方法 显示英文标题

标题： An entropy stable discontinuous Galerkin method for the two-layer shallow water equations on curvilinear meshes

摘要： 我们提出了一种熵稳定节点不连续伽辽金谱元方法（DGSEM），用于二维曲线网格上的双层浅水方程。 我们在半离散层面上模仿连续熵分析，该DGSEM构建在勒让德-高斯-洛巴托（LGL）节点上。 使用LGL节点使集中节点DGSEM具有求和-by-parts性质，这是离散分析中的关键。 该近似利用体积贡献的等效通量差分形式，生成控制方程的熵守恒分裂形式。 然后应用一种特定的熵守恒数值表面通量与非守恒项的离散化组合，以获得高阶路径守恒格式，该格式在不连续地形情况下具有熵守恒和良好平衡特性。 在界面处添加耗散以创建满足离散情况下热力学第二定律的熵稳定近似。 我们通过数值测试验证了理论结果，并展示了关于方案收敛性、熵稳定性和良好平衡性的结果。 显示英文摘要

摘要： 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.