Constraint preserving boundary conditions for the Ideal Newtonian MHD equations

Magnetic fields play an important role in the behavior of plasmas and are thought mediate important effects like dynamos in the core of planets and the formation of jets in active galactic nuclei and gamma ray bursts; induce a variety of magnetic instabilities; realize solar flares, etc. (see e.g. [1,2]). Understanding the role of magnetic fields in these and other phenomena have spurred through the years many efforts to obtain solutions of the magnetohydrodynamic equations. The non-linear nature of these equations limits the understanding that can be gained in a particular problem via analytical techniques. This implies that solutions for complex systems must be obtained by numerical means and a suitable numerical implementation must be constructed for this purpose. Such implementation must be able to evolve the solution to the future of some initial configuration and guarantee its quality. A delicate, subsidiary quantity, can be monitored in part to estimate this. This quantity is the “monopole constraint” ∂iB i which must be zero at the analytical level for a consistent solution. This quantity is not a part of the main variables, rather it is a derived quantity which should be satisfied by a true physical solution. In practice, unless a numerical implementation of the MHD equations is carefully designed this constraint can be severely violated. This, in turn, signals (and is sometimes the cause) of a degrading numerical solution. For these reasons, several approaches have been investigated and developed for guaranteeing a controlled behavior of this quantity. One such approach is known as the constraint transport technique [3,4] which adopts a particular algorithm that staggers the variables appropriately to ensure the satisfaction of the constraint at round-off level within Finite Difference and Finite Elements techniques. This approach has been quite successful in a number of applications across different disciplines and particularly relevant in astrophysics applications [5,6,7,8,9,10,11,12]. However, by design it imposes limits on the algorithmic options available to an implementation. This fact can be at odds, or introduce complications, with applications where adaptive mesh refinement is required and/or advanced numerical techniques (that exploit useful properties of the equations) are adopted. An alternative approach, which controls the constraint at truncation-error level maintains complete freedom in the numerical techniques to be adopted. This approach, referred to as divergence cleaning puts the burden to control the constraint not on the algorithm to be employed