We present a discontinuous Galerkin method for moist atmospheric dynamics, with and without warm rain. By considering a combined density for water vapour and cloud water, we avoid the need to model and compute a source term for condensation. We recover the vapour and cloud densities by solving a pointwise non-linear problem each time step. Consequently, we enforce the requirement for the water vapour not to be supersaturated implicitly. Together with an explicit time-stepping scheme, the method is highly parallelisable and can utilise high-performance computing hardware. Furthermore, the discretisation works on structured and unstructured meshes in two and three spatial dimensions. We illustrate the performance of our approach using several test cases in two and three spatial dimensions. In the case of a smooth, exact solution, we illustrate the optimal higher-order convergence rates of the method.

翻译：我们提出了一种适用于湿大气动力学的间断伽辽金方法，涵盖有暖雨和无暖雨两种情况。通过考虑水汽和云水的联合密度，我们避免了模拟和计算凝结源项的必要性。我们通过每一步求解一个逐点非线性问题来恢复水汽和云水密度。因此，我们隐式地满足了水汽不出现过饱和的要求。结合显式时间推进格式，该方法具有高度并行性，能够利用高性能计算硬件。此外，该离散化方法适用于二维和三维空间中的结构化与非结构化网格。我们通过若干二维和三维空间中的测试案例展示了该方法的性能。在具有光滑精确解的情况下，我们说明了该方法的最优高阶收敛速度。