To circumvent the scalability problem due to global communication involved in spectral transforms, a vertical-slice version of the dynamical core where all calculations are performed in grid-point space has been built for AROME, Météo-France's operational limited-area model. It is shown in an idealized but nevertheless physically relevant framework that, despite this major change, it is possible to keep the other main characteristics of the model (constant-coefficient semi-implicit scheme, semi-Lagrangian transport scheme, A-grid, mass-based coordinate, etc.). A Krylov solver is used to solve the implicit problem. Using the solution given by the spectral model as a reference in terms of quality and required accuracy in an operational context, the chosen parameters of the Krylov solver are carefully tuned to maximize its convergence speed. The Krylov solver consists of several applications of a sparse operator, whose stencil is similar to that of the operator applied during the small time step of split-explicit schemes traditionally used in HEVI (Horizontally Explicit/Vertically Implicit) models.HEVI models are considered particularly scalable in the current parallelization paradigm and are used as a reference for scalability in this study. Results show that a fast convergence can be reached, such that only a few applications of the sparse operator are required. This suggests an improvement in scalability compared to the spectral version. Experiments have been formulated regardless of the future strategy considered for temporal and spatial resolutions, i.e., increasing spatial resolution while either maintaining a high Courant number or reducing it.