“…More recently, Maire proposed a cell-centred Lagrangian formulation to capture axisymmetrical, that is, spherical, imploding shock waves [11]; Illenseer and Duschl generalized the two-dimensional central upwind to curvilinear grids [12]; Clain and collaborators extended a multi-slope MUSCL scheme to cylindrical coordinates [13]. In the framework of Lagrangian or ALE solvers, the explosion and implosion problems in two spatial dimensions were solved by Cheng and Shu using a cell-centred Lagrangian scheme [14]; Loubère and others used the reconnection algorithm for ALE framework in cylindrical coordinates [15].The present approach moves from the mixed FV/finite element (FE) approach [16], which has been already successfully applied to the multidimensional Cartesian case [17,18] and the cylindrical case with axial symmetry [19], within the throughflow approximation [20] and more recently, in a unified fashion form for orthogonal coordinate systems [21]. To link these two different numerical schemes, it is necessary to introduce suitable equivalence conditions that relate the FV metrics, that is, cell volumes and integrated normals, to the FE integrals.…”