Well-balanced and scale-dependent time integration and for weakly compressible (atmospheric) flows
Atmospheric flows feature a cascade of characteristic scales induced by the presence of several independent small dimensionless parameters in the governing equations. The most important ones are the Mach, Froude, and Rossby numbers. Depending on the length and time scales considered, different asymptotic limit regimes prevail, leading to very different typical flow behavior, as revealed by single-scale asymptotic analysis. The development of numerical schemes for the full three-dimensional compressible flow equations that properly respect such balances in each individual regime has a considerable history.
Current super-computers allow atmospheric modellers to resolve a broad range of these scales in one and the same simulation. The numericist is thus challenged to devise numerical integrators that simultaneously respect the asymptotic balances across all length and time scale combinations that arise in a particular flow case.
In the first part of this presentation I will introduce the asymptotic characterization of scale-dependent atmospheric flow regimes. In the second part, I discuss recent developments towards related multi-scale time integrators.