This thesis regards the numerical simulation of inviscid compressible ideal gases which are described
by the Euler equations. We propose a novel implicit explicit (IMEX) relaxation scheme to simulate
flows from compressible as well as near incompressible regimes based on a Suliciu-type relaxation
model. The Mach number plays an important role in the design of the scheme, as it has great
influence on the flow behaviour and physical properties of solutions of the Euler equations. Our
focus is on an accurate resolution of the Mach number independent material wave. A special feature
of our scheme is that it can account for the influence of a gravitational field on the fluid flow and is
applicable also in small Froude number regimes. The time step of the IMEX scheme is constrained
only by the eigenvalues of the explicitly treated part and is independent of the Mach number allowing
for large time steps independent of the flow regime. In addition, the scheme is provably asymptotic
preserving and well-balanced for arbitrary a priori known hydrostatic equilibria independently of
the considered Mach and Froude regime. Also, the scheme preserves the positivity of density and
internal energy throughout the simulation, it is well suited for physical applications. To increase the
accuracy, a natural extension to second order is provided. The theoretical properties of the given
schemes are numerically validated by various test cases performed on Cartesian grids in multiple
space dimensions