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tokens: panel: p rich ). For the helium burning problem, discussed in this study, we find global agreement between the two solvers, whereas there are larger differences in the small-scale features. Figure 1. Density colour map of the in initial state for the floating torus. Figure 2. Density colour map of the floating torus at time 0.2 sec (left-hand panel: vulcan ; right-hand panel: rich) . Figure 3. Density colour map of the floating torus at time 0.3 sec (left-hand panel: vulcan ; right-hand panel: rich) . In particular, although the two solvers successfully suppress numerical transition to detonation in very different ways, the final conclusions are the same. Finally, some caution must be taken due to the issue of possible turbulence. We recognize that our simulations cannot follow fully developed turbulence since the spatial resolution is too coarse and also the missing third dimension limits the correct cascading of turbulent scales. It is well known that fully developed turbulence may enhance burning and as a result may trigger detonations. However, since we reach detonations in our 2D simulations, the presence of possible turbulence cannot alter our final conclusions. 2.1 The <sc>vulcan</sc> methodology The code v2d (Vulcan-2D; Livne 1993 ) consists of an ALE method with a second-order advection scheme. In the problems discussed here, the grid moves rather slowly and the simulations are nearly Eulerian. For astrophysical combustion problems, a ‘multi-phase’ method has been added, which allows to handle separately fuel and ashes as different substances. The advection algorithm can use either a ‘slopes method’ (a la Van-Leer scheme) or VOF (Volume of Fluid) method. VOF is actually a family of schemes in which sharp boundaries between phases are constructed for advecting phases with minimal numerical diffusion (Hirt 1981 ; Ubbnik 1999 ; Gopala 2008 ). In v2d , we implement the basic idea during the advection step, by constructing, in each mixed cell, a local planar interface using the known partial volume fractions. For two phases, the partial volumes and the orientation of the surface provide enough information for interface reconstruction. The orientation, or a vector normal to the interface, is calculated from the phases gradients, based on cell centres values. In cases of more than two phases, the algorithm is much more complicated, not uniquely defined and less accurate. Once we have the local linear interface, we use it to cut the boundaries of the cell and to compute exact phase fluxes between cells. The VOF method is rather accurate as long as the boundaries between the phases are smooth. When small-scale fragmentation develops it is harder to define accurate normals and as a result the method becomes less reliable. The VOF method is more appropriate to the issue of this work and was implemented in all our helium burning simulations. The phases are defined according to a ‘critical temperature,’ which is in our case T c = 3 × 10 8 K.