Modellierung und Approximation feuchter atmosphärischer Strömungen unter Berücksichtigung topographischer Effekte

在流体力学中，守恒性质引起了极大的关注。在存在湿过程时，相边界可以用不连续来模拟。因此，一个数学上合适的公式是弱形式的质量、动量和能量守恒定律。在气象模拟中，存在许多可供选择的公式，这些公式允许以这样或那样的方式处理不连续的简单离散。在这个项目中，我们比较了质量守恒和动量守恒两种不同的描述热力学过程的方法。我们考虑并比较了两种不同类型的有限体积方法：波传播方法（其中数值通量基于局部黎曼问题的解）和传统上用于数值天气预报（NWP）的方法（其中每个分量的通量通过插值获得）。为了解决有成功前景的气象问题，数值方法应该能够处理地形、小马赫数和弗劳德数，以及硬相变过程。我们在笛卡尔网格上使用离散，包括嵌入边界的切割细胞。

In fluid dynamics conservation properties attract a great deal of attention. In the presence of moist processes phase boundaries can be modelled by discontinuities. Therefore a mathematically appropriate formulation are the conservation laws for mass, momentum and energy in weak form. In meteorological modelling there exist numerous alternative formulations which allow for simple discretisations taking care of discontinuities in one way or another. In this project we compare mass- and momentum conserving methods which differ in their description of thermodynamic processes. We consider and compare two different classes of finite volume methods: wave-propagation methods (where the numerical flux is based on the solution of local Riemann problems) and methods which are traditionally used in numerical weather prediction (NWP) (where in each component the flux is obtained by interpolation). To address meteorological problems with some prospect of success numerical methods should be able to cope with orography, small Mach- and Froude numbers, as well as with stiff phase transition processes. We use discretisations on Cartesian grids including cut cells at embedded boundaries.