Positivity-preserving and symmetry-preserving Lagrangian schemes for compressible Euler equations in cylindrical coordinates

For a Lagrangian scheme defined in the cylindrical coordinates, two important issues are whether the scheme can maintain spherical symmetry (symmetry-preserving) and whether the scheme can maintain positivity of density and internal energy (positivity-preserving). While there were previous results in the literature either for symmetry-preserving in the cylindrical coordinates or for positivity-preserving in cartesian coordinates, the design of a Lagrangian scheme in cylindrical coordinates, which is high order in one-dimension and at least second order in two-dimensions, and can maintain both spherical symmetry-preservation and positivity-preservation simultaneously, is challenging. In this paper we design such a Lagrangian scheme and provide numerical results to demonstrate its good behavior.