Simulation of magnetohydrodynamic flows - A constrained transport method

We discuss an optimal strategy for treating the magnetohydrodynamic (MHD) field transport (induction) equation. The induction equation is shown to assume its simplest form when written in terms of the contravariant forms of velocity and magnetic vector density. The approach places the induction equation in integral form and uses the magnetic flux as a fundamental variable. We describe a new numerical technique, called constrained transport (CT), for evolving the induction equation in a way that maintains vanishing divergence of the poloidal (constrained) field components to within machine round-off error. The scheme places no restrictions on the order of accuracy of the transport technique and is shown to work in three dimensions as well as in two. We show that CT allows incorporation of an adaptive mesh algorithm in a natural way. No barrier is evident to prevent the use of the technique in nonideal MHD simulations. Some past treatments of the MHD induction equation are reviewed, with attention to the question of constraint satisfaction. We show how the 3 + 1 formalism of numerical relativity can be used to express the general relativistic MHD equations in a form suitable for numerical evolution. The relativistic form of the magnetic induction equation is then shown to fit naturally within the CT scheme. An axisymmetric, two-dimensional finite-difference code has been constructed incorporating the new method for the induction equation. Results obtained using the CT method are compared with those obtained from a code employing vector potential evolution. Several models involving magnetized flow near a black hole are used as calibrations of the CT scheme.