  Superbursts are rare and energetic thermonuclear carbon flashes observed to
occur on accreting neutron stars. We create the first multi-zone models of
series of superbursts using a stellar evolution code. We self-consistently
build up the fuel layer at different rates, spanning the entire range of
observed mass accretion rates for superbursters. For all models light curves
are presented. They generally exhibit a shock breakout, a precursor burst due
to shock heating, and a two-component power-law decay. Shock heating alone is
sufficient for a bright precursor, that follows the shock breakout on a short
dynamical time scale due to the fall-back of expanded layers. Models at the
highest accretion rates, however, lack a shock breakout, precursor, and the
first power law decay component. The ashes of the superburst that form the
outer crust are predominantly composed of iron, but a superburst leaves a
silicon-rich layer behind in which the next one ignites. Comparing the model
light curves to an observed superburst from 4U 1636-53, we find for our
accretion composition the best agreement with a model at three times the
observed accretion rate. We study the dependence on crustal heating of
observables such as the recurrence time and the decay time scale. It remains
difficult, however, to constrain crustal heating, if there is no good match
with the observed accretion rate, as we see for 4U 1636-53.
