Non-Oscillatory Central Schemes for One- and Two-Dimensional MHD Equations

In this paper we utilize a family of high-resolution, non-oscillatory central schemes for the approximate solution of the equations of ideal magnetohydrodynamics (MHD) in oneand two-space dimensions. We present several prototype problems. Solutions of one-dimensional shock-tube problems is carried out using secondand third-order central schemes [19, 18], and we use the second-order central scheme [11] which is adapted for the solution of the two-dimensional Kelvin-Helmholtz and Orszag-Tang problems. A qualitative comparison reveals an excellent agreement with previous results based on upwind schemes. Central schemes, however, require little knowledge about the eigen-structure of the problem — in fact, we even avoid an explicit evaluation of the corresponding Jacobians, while at the same time they eliminate the need for dimensional splitting. The oneand two-dimensional computations reported in this paper demonstrate the remarkable versatility of central schemes as black-box, Jacobian-free MHD solvers. AMS subject classification: Primary 65M10; Secondary 65M05