  We use hydrodynamical simulations in a $(256\;{\rm pc})^3$ periodic box to
model the impact of supernova (SN) explosions on the multi-phase interstellar
medium (ISM) for initial densities $n = 0.5-30$ cm$^{-3}$ and SN rates $1-720$
Myr$^{-1}$. We include radiative cooling, diffuse heating, and the formation of
molecular gas using a chemical network. The SNe explode either at random
positions, at density peaks, or both. We further present a model combining
thermal energy for resolved and momentum input for unresolved SNe. Random
driving at high SN rates results in hot gas ($T\gtrsim 10^6$ K) filling $> 90$%
of the volume. This gas reaches high pressures ($10^4 < P/k_\mathrm{B} < 10^7$
K cm$^{-3}$) due to the combination of SN explosions in the hot, low density
medium and confinement in the periodic box. These pressures move the gas from a
two-phase equilibrium to the single-phase, cold branch of the cooling curve.
The molecular hydrogen dominates the mass ($>50$%), residing in small, dense
clumps. Such a model might resemble the dense ISM in high-redshift galaxies.
Peak driving results in huge radiative losses, producing a filamentary ISM with
virtually no hot gas, and a small molecular hydrogen mass fraction ($\ll 1$%).
Varying the ratio of peak to random SNe yields ISM properties in between the
two extremes, with a sharp transition for equal contributions. The velocity
dispersion in HI remains $\lesssim 10$ km s$^{-1}$ in all cases. For peak
driving the velocity dispersion in H$_\alpha$ can be as high as $70$ km
s$^{-1}$ due to the contribution from young, embedded SN remnants.
