Description: Convection-permitting limited-area models based on the same spectral semi-implicit semi-Lagrangian (SL) techniques which are used in the ECMWF global model, are run operationally in several countries of the ALADIN/HIRLAM consortium. Forecasters reported a general tendency for these models to produce overestimated precipitation and unrealistic divergent winds at the edges of the cold outflows generated by the precipitation evaporation in the vicinity of the convective clouds. These grid-point storms have been associated with a spurious behaviour of the pointwise interpolation used in the semi-Lagrangian scheme, where grid-scale buoyant updrafts create strong small scale convergence near the surface. A modification of the interpolation weights in the SL transport scheme introduces the concept of cell-averaging into the traditional pointwise SL scheme which improves the conservation property of the scheme and eliminates the spurious mode. The COMAD (for COntinuous Mapping about Departure points) correction applied to the standard interpolation weights takes into account the deformation of the air parcels along each direction of interpolation in order to improve the continuity and the conservative property of the re-mapping between the model grid-points and the origin points of the backward trajectories. The method has been validated with the small planet configuration of the Integrated Forecast System (IFS) at ECMWF and with the limited area version of the same dynamics used for the AROME (Meteo-France) and HARMONIE (HIRLAM) models. The pathological behaviour of grid-scale buoyant flows permitted by these dynamical cores is corrected by the COMAD interpolations. The precipitation forecast in the convection-permitting models AROME/HARMONIE which shows an overestimation of intense convective precipitation is systematically improved by the new weights.