A Discontinuous Galerkin (DG) method for the solution of the shallow-water equations (SWE) on arbitrary 2-dimensional (2D) manifolds is presented. To this purpose the SWE are formulated in covariant form using tensor notation. This allows to correctly transform the numerical fluxes between the local coordinate systems of any two neighbouring grid cells. In particular, the covariant form of the numerical diffusion term in the Lax-Friedrichs numerical flux has been derived, too. This general approach has the advantage that it avoids any coordinate singularity. It is tested for the SWE on the sphere with several standard test setups. Beyond this, a recently published test case with an analytic solution for linear inertial-gravity wave expansion has been performed. The derived formalism on arbitrary 2D manifolds allows an easy extension from the sphere to the ellipsoid. The comparison of a barotropic instability test case for the earth shows a non-negligible difference between the solution on these two bodies. The presented approach may be a starting point for the development of a dynamical core for numerical weather and climate prediction models based on the DG method.

提出了一种在任意二维（2D）流形上求解浅水方程（SWE）的不连续伽勒金（DG）方法。为此，使用张量表示法将SWE配制成协变形式。这允许正确转换任意两个相邻网格单元的局部坐标系之间的数值通量。特别地，也已经导出了Lax-Friedrichs数值通量中数值扩散项的协变形式。这种通用方法的优点是避免了任何坐标奇异性。已通过几种标准测试设置对球上的SWE进行了测试。除此之外，还执行了一个最近发布的带有线性惯性重力波扩展解析解决方案的测试案例。在任意2D流形上派生的形式主义可以轻松地从球体扩展到椭圆体。对地球的正压不稳定测试案例的比较表明，在这两个物体上的解决方案之间的差异不可忽略。所提出的方法可能是开发基于DG方法的数值天气和气候预测模型动态核心的起点。