Simulation of longitudinal Landau damping in bunches with space charge

For a single hadron bunch affected by longitudinal space charge in a stationary rf bucket we analyze the frequency spectrum close to the expected loss of Landau damping for the lowest order dipole mode. For different bunch intensity parameters we obtain the bunch oscillation spectrum from a conventional longitudinal particle tracking code with a grid-based space charge solver. We validate selected results against a grid-less space charge solver. We highlight the importance of the choice of the cut-off parameter $h_c$ in the space charge impedance for the long-term accuracy of grid-based schemes. For typical bunch parameters in an ion synchrotron at injection energies we find that the branching point, where the dipole mode frequency emerges from the incoherent synchrotron frequency spectrum, as well as the damping of the dipole mode do not depend on $h_c$, chosen well below the actual value for realistic beam pipes.