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The present study proposes a highly accurate lattice Boltzmann direct coupling cell-vertex algorithm, well suited for industrial purposes, making it highly valuable for aeroacoustic applications. It is indeed known that the convection of vortical structures across a grid refinement interface, where cell size is abruptly doubled, is likely to generate spurious noise that may corrupt the solution over the whole computational domain. This issue becomes critical in the case of aeroacoustic simulations, where accurate pressure estimations are of paramount importance. Consequently, any interfering noise that may pollute the acoustic predictions must be reduced. The proposed grid refinement algorithm differs from conventionally used ones, in which an overlapping mesh layer is considered. Instead, it provides a direct connection allowing a tighter link between fine and coarse grids, especially with the use of a coherent equilibrium function shared by both grids. Moreover, the direct coupling makes the algorithm more local and prevents the duplication of points, which might be detrimental for massive parallelization. This work follows our first study (Astoul~\textit{et al. 2020}) on the deleterious effect of non-hydrodynamic modes crossing mesh transitions, which can be addressed using an appropriate collision model. The Hybrid Recursive Regularized model is then used for this study. The grid coupling algorithm is assessed and compared to a widely-used cell-vertex algorithm on an acoustic pulse test case, a convected vortex and a turbulent circular cylinder wake flow at high Reynolds number.