The main source of difficulties one comes across when numerically solving compressible Euler and shallow water equations is lack of  smoothness as their solutions may develop very complicated nonlinear wave structures. The level of complexity may increase even further when solutions reveal a multiscale character and/or the system include additional source terms such as Coriolis forces. In such cases, it is extremely important to design a numerical method which is not only consistent with the studied systems, but also preserves certain structural and asymptotic properties of the underlying problems at the discrete level.

In this talk, I will discuss recent advances in the development of AP schemes that provide accurate and efficient numerical solutions in low Mach/Froude number asymptotic regimes.