Abstract: The aim of the present talk is two-fold. We will firstly introduce a second order IMEX finite volume method in order to approximate efficiently the Euler equations with the gravity source term that are based on the so-called acoustic/advection splitting strategy. More precisely, we split the whole nonlinear system of the Euler equations into a stiff linear part governing fast acoustic and gravity waves and a non-stiff nonlinear part that models slow nonlinear advection effects. For time discretization we have used second order globally stiff fly accurate IMEX scheme and approximate stiff linear operator implicitly and the non-stiff nonlinear operator explicitly. We will prove that this IMEX FVM is asymptotic preserving, i.e. it is stable and consistent uniformly with respect to the Mach number.

In the second part, we study propagation of uncertainties due to data or model parameters and apply the generalized polynomial chaos. This leads to the stochastic Galerkin method. Thus, the stochastic space is approximated by the spectral Galerkin method, whereas the space-time domain is approximated by the above IMEX FVM. We illustrate the application of the uncertainty quantification techniques for some atmospheric flows. This work has been done in collaboration with A. Kurganov (SUSTech), A. Chertock (Raleigh), and B. Wiebe, L. Yelash and P. Spichtinger (Mainz).