We present a novel discontinuous Galerkin finite element method for numerical simulations of the rotating thermal shallow water equations in complex geometries using curvilinear meshes, with arbitrary accuracy. We derive an entropy functional which is convex, and which must be preserved in order to preserve model stability at the discrete level. The numerical method is provably entropy stable and conserves mass, buoyancy, vorticity, and energy. This is achieved by using novel entropy stable numerical fluxes, summation-by-parts principle, and splitting the pressure and convection operators so that we can circumvent the use of chain rule at the discrete level. Numerical simulations on a cubed sphere mesh are presented to verify the theoretical results. The numerical experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilisation.

翻译：我们提出了一种新颖的间断伽辽金有限元方法，用于在复杂几何区域中使用曲线网格对旋转热浅水方程进行数值模拟，并达到任意精度。我们推导了一个凸的熵泛函，该泛函必须在离散层面上被保持以维持模型稳定性。该数值方法在理论上被证明是熵稳定的，并且守恒质量、浮力、涡度和能量。这是通过使用新颖的熵稳定数值通量、求和-分部原则以及对压力和对流算子进行分裂来实现的，从而可以在离散层面上避免链式法则的使用。我们给出了在立方球网格上的数值模拟以验证理论结果。数值实验表明，该方法在充分发展的湍流状态下具有鲁棒性，能在无需任何耗散的情况下稳定运行。熵稳定通量足以控制由地转湍流产生的网格尺度噪声，从而消除了对人工稳定化的需求。