We design and analyse an energy stable, structure preserving and well-balanced scheme for the Ripa system of shallow water equations. The energy stability of the numerical solutions is achieved by introducing appropriate stabilisation terms in the discretisation of the convective fluxes of mass and momenta, the pressure gradient and the topography source term. A diligent choice of the interface values of the water height and the temperature ensures the well-balancing property of the scheme for three physically relevant hydrostatic steady states. The explicit in time and finite volume in space scheme preserves the positivity of the water height and the temperature, and it is weakly consistent with the continuous model equations in the sense of Lax-Wendroff. The results of extensive numerical case studies on benchmark test problems are presented to confirm the theoretical findings.

翻译：本文设计并分析了一种针对浅水方程Ripa系统的能量稳定、结构保持且平衡的数值格式。通过在质量与动量对流通量、压力梯度及地形源项的离散化中引入适当的稳定项，实现了数值解的能量稳定性。对水深和温度界面值的审慎选择，确保了该格式对三种物理相关的静水稳态具有平衡特性。这种时间显式、空间有限体积的格式保持了水深和温度的正性，并在Lax-Wendroff意义下与连续模型方程弱相容。文中通过大量基准测试问题的数值算例结果，验证了理论结论。