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

A complete set of equivalence conditions, connecting the node-centered finite volume and the mass-lumped finite element schemes, is derived for the first time for the one-dimensional compressible Euler equations with cylindrical and spherical symmetry. Analytical expressions for the evaluation of the equivalent cell volumes and interface normals in terms of the finite element integrals are presented. Numerical experiments for compressible unsteady flows, including expanding and converging shock problems, are carried out using the new approach and the differences with the results from a finite volume scheme violating the equivalence conditions are discussed.