A finite volume implicit time integration method for solving the equations of ideal magnetohydrodynamics for the hyperbolic divergence cleaning approach

A finite volume numerical technique is proposed to solve the compressible ideal MHD equations for steady and unsteady problems based on a quasi-Newton implicit time integration strategy. The solenoidal constraint is handled by a hyperbolic divergence cleaning approach allowing its satisfaction up to machine accuracy. The conservation of the magnetic flux is computed in a consistent way using the numerical flux of the finite volume discretization. For the unsteady problem, the time accuracy is obtained by a Newton subiteration at each physical timestep thereby converging the solenoidal constraint to steady state. We perform extensive numerical experiments to validate and demonstrate the capabilities of the proposed numerical technique.