Multi-dimensional numerical code to solve the equations for magnetohydrodynamics (MHD) is developed. It is based on the algorithm called the Total Variation Diminishing(TVD) scheme, which is a second order extension of the Roe type upwind scheme. Strang-type operator splitting is used to treat multiple spatial dimensions. The constraint of divergence-free field is ensured explicitly by correction via gauge transformation in each time step. Diffusion is added to the solution of the ideal MHD equations. Alternating direction implicit(ADI) method is adopted to solve the diffusion equations.
The code is tested by reproducing the evolution of the Orszag-Tang vortex in compressible media. The test confirms that the code well represents the features involved in turbulent MHD flows. We have captured the MHD turbulence with the hydrodynamic turbulence. It is seen that the compressibility is an important factor in determining the nature of the turbulence. We have also studied the dependence of the vortex evolution on the plasma β.
