In the present paper, we propose another space discretization of the kinematic constraint for a single grid multi velocity formulation leading to a better approximation of the interface physics. The paper is organized as follows. In the next section, we will briefly review the problem to solve and the single mesh multi-velocity discretization strategy. In Section 3, we will then introduce our new cell-based discretization of the normal velocity continuity constraint on the interface, develop the proper velocity enrichment, and analyse its stability. A generic time discretization is next introduced in Section 4 and adapted to the specific space discretization proposed herein. The proposed methodology is finally validated on a few numerical examples.