Hyperbolic Divergence Cleaning for the MHD Equations

In simulations of magnetohydrodynamic (MHD) processes the violation of the divergence constraint causes severe stability problems. In this paper we develop and test a new approach to the stabilization of numerical schemes. Our technique can be easily implemented in any existing code since there is no need to modify the solver for the MHD equations. It is based on a modified system in which the divergence constraint is coupled with the conservation laws by introducing a generalized Lagrange multiplier. We suggest a formulation in which the divergence errors are transported to the domain boundaries with the maximal admissible speed and are damped at the same time. This corrected system is hyperbolic and the density, momentum, magnetic induction, and total energy density are still conserved. In comparison to results obtained without correction or with the standard “divergence source terms,” our approach seems to yield more robust schemes with significantly smaller divergence errors.