  While feedback from Active Galactic Nuclei (AGN) is an important heating
source in the centre of galaxy clusters, it is still unclear how the feedback
energy is injected into the intracluster medium (ICM) and what role different
numerical approaches play. Here, we compare four hydrodynamical schemes in
idealized simulations of a rising bubble inflated by AGN feedback in a hot
stratified ICM: (traditional) smoothed particle hydrodynamics (TSPH), a
pressure flavour of SPH (PSPH), a meshless finite mass (MFM) scheme, as well as
an Eulerian code with adaptive mesh refinement. In the absence of magnetic
fields, the bubble is Kelvin-Helmholtz unstable on short enough time scales to
dissolve it fully in the ICM, which is captured by MFM and RAMSES simulations,
while in the TSPH simulation the bubble survives. When the ICM is turbulent,
mixing of the bubble with the ICM is accelerated. This occurs if the numerical
scheme can capture the instabilities well. The differences in the evolution of
the bubble has a surprisingly small influence on the thermal structure of the
ICM. However, in the simulations with MFM and RAMSES the bubble disruption
leads to turbulent stirring of the ICM which is suppressed in SPH. In the
latter the thermal energy remains trapped in the bubble and is transported to
large radii. We discuss if the choice of hydrodynamical schemes can lead to
systematic differences in the outcomes of cosmological simulations.
