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Aims. This work presents a new hydrodynamical algorithm to studyastrophysical detonations. A prime motivation of this development is thedescription of a carbon detonation in conditions relevant to superbursts, whichare thought to result from the propagation of a detonation front around thesurface of a neutron star in the carbon layer underlying the atmosphere.Methods. The algorithm we have developed is a finite-volume method inspired bythe original MUSCL scheme of van Leer (1979). The algorithm is of second-orderin the smooth part of the flow and avoids dimensional splitting. It is appliedto some test cases, and the time-dependent results are compared to thecorresponding steady state solution. Results. Our algorithm proves to be robustto test cases, and is considered to be reliably applicable to astrophysicaldetonations. The preliminary one-dimensional calculations we have performeddemonstrate that the carbon detonation at the surface of a neutron star is amultiscale phenomenon. The length scale of liberation of energy is $10^6$ timessmaller than the total reaction length. We show that a multi-resolutionapproach can be used to solve all the reaction lengths. This result will bevery useful in future multi-dimensional simulations. We present alsothermodynamical and composition profiles after the passage of a detonation in apure carbon or mixed carbon-iron layer, in thermodynamical conditions relevantto superbursts in pure helium accretor systems.

Hydrodynamical simulation of detonations in superbursts. I. The hydrodynamical algorithm and some preliminary one-dimensional results