[en] The induction linac bunches of heavy-ion-fusion scenarios are strongly influenced by the longitudinal space-charge impedance. This is in distinct contrast to relativistic bunches in storage rings where most of the data on stability have been obtained. Simulation results reveal that when space-charge effects are large, the stability requirement of small growth rate relative to the synchrotron frequency for relativistic bunches is replaced by the relaxed condition of small growth rate relative to the frequency spacing of the space charge wave modes on the bunch. Dispersive effects from finite pipe size tend to make the lower frequencies less susceptible to instability than higher frequencies. Since induction modules have a high resistive component only for the lowest bunch modes, stability is better than would occur for a broadband impedance of comparable magnitude. These results indicate that long-term longitudinal bunch stability is realizable for induction-linac drivers for heavy-ion fusion