  On the basis of the recent group classification of the one-dimensional
magnetohydrodynamics (MHD) equations in cylindrical geometry, the construction
of symmetry-preserving finite-difference schemes with conservation laws is
carried out. New schemes are constructed starting from the classical completely
conservative Samarsky-Popov schemes. In the case of finite conductivity,
schemes are derived that admit all the symmetries and possess all the
conservation laws of the original differential model, including previously
unknown conservation laws. In the case of a frozen-in magnetic field (when the
conductivity is infinite), various schemes are constructed that possess
conservation laws, including those preserving entropy along trajectories of
motion. The peculiarities of constructing schemes with an extended set of
conservation laws for specific forms of entropy and magnetic fluxes are
discussed.

Fluid Dynamics;Numerical Analysis;Numerical Analysis