Comparison of several finite difference methods for magnetohydrodynamics in 1D and 2D

The comparison of several finite difference methods for ideal magnetohydrodynamics (MHD) is presented. Compared finite difference methods include composite schemes, central scheme, WENO, component wise CWENO, and public freely available packages Nirvana and Flash. ID Cartesian tests concern smooth, Brio-Wu and intermediate shock formation problems. From 2D Cartesian tests we shortly present Orszag-Tang vortex problem and shock-cloud interaction problem. As we are interested in the generalization of schemes from Cartesian to cylindrical r - z geometry, we include also generalization of composite and CWENO schemes to cylindrical geometry with their application to 2D conical z-pinch problem.