•Compressible simulations of near-wall collapses.•Rebound and pressure impact on the wall are captured.•Comprehensive study of the central mechanism for induced pressure peaks on the wall.•Highest grid resolution for 3D simulations of bubble collapses.•Grid study using six different grid resolutions and considering various stand-off distances. We consider the collapse behavior of cavitation bubbles near walls under high ambient pressure conditions. Generic configurations with different stand-off distances are investigated by numerical simulation using a fully compressible two-phase flow solver including phase change. The results show that the stand-off distance has significant effects on collapse dynamics, micro-jet formation, rebound, and maximum wall pressure. A relation between cavitation induced material damage and corresponding collapse mechanisms is obtained from pressure-impact data at the wall. We analyze the resolution dependence of collapse and rebound and the observed maximum pressure distributions. The comparison of the results on six different grid resolutions shows that main collapse features are already captured on the coarsest resolution, while the peak pressures are strongly resolution dependent.