SPH in spherical and cylindrical coordinates

New kernel functions for spherically, planar and cylindrically symmetric problems are developed, based on the fundamental interpolation theory of SPH. The Lagrangian formalism is used to derive the corresponding set of modified SPH equations of motion. The results show good agreement both with analytical and numerical results, in the case of the Sod shock tube test, the Noh infinite shock problem, and the Sedov point explosion test. The formulation has also been included in a 3D cylindrically symmetric problem of two colliding spherical shocks. For this latter problem, the results are presented allowing both a constant and a variable resolution. The results clearly demonstrate the capability of the new formulation to solve the singularity problem at the symmetry axis.