On the relation between finite element and finite volume schemes for compressible flows with cylindrical and spherical symmetry

Equivalence conditions between linear Lagrangian finite element and node‐centred finite volume schemes for conservation laws in cylindrical coordinates

“…The resulting numerical scheme is formally second-order accurate both in space and time for smooth solution, as it has been shown in the two-dimensional axisymmetric formulation [19] or the one-dimensional spherical and cylindrical one [21]. In the presence of shock waves, the scheme is first-order accurate in space.…”

“…To this purpose, the one-dimensional versions of the scheme, presented in details in [21], are first evaluated against the analytical reference solution due to Guderley [33] for the motion of a converging shock wave in both the cylindrical (R Â) and spherical (Z R) symmetric cases. To this purpose, the one-dimensional versions of the scheme, presented in details in [21], are first evaluated against the analytical reference solution due to Guderley [33] for the motion of a converging shock wave in both the cylindrical (R Â) and spherical (Z R) symmetric cases.…”

“…Equivalence conditions are used to devise a class of FV schemes, in which all grid-dependent quantities are defined in terms of FE integrals. 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]. The two-dimensional schemes for the polar and the axisymmetrical cases are also explicitly derived.…”

“…The construction of an equivalent FV scheme for the Euler equation for one-dimensional problems endowed with cylindrical symmetry is discussed in [21]. The construction of an equivalent FV scheme for the Euler equation for one-dimensional problems endowed with cylindrical symmetry is discussed in [21].…”

“…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.…”