A new hydrodynamical algorithm to study astrophysical detonations is presented. A prime motivation of this development is the description of a carbon detonation in conditions relevant to superbursts, which are thought to result from the propagation of a detonation front around the surface of a neutron star in the carbon layer underlying the atmosphere. The one-dimensional calculations we have performed demonstrate that the carbon detonation at the surface of a neutron star is a multiscale phenomenon. We show that a multi-resolution approach can be used to solve all the reaction lengths. For mixed H/He accreting systems, we have introduced a new reduced network to study the impact of the photodesintegration of the heavy elements on the detonation.