Constraint preserving boundary conditions for the Ideal Newtonian MHD equations

CÉCERE, M. A.; LEHNER, L.; REULA, O.

La Plata, Buenos Aires, Argentina

Workshop; First La Plata Internacional School on Astronomy and Geophysics, "Compact objects and their emission"; 2008

Grupo de Astrofísica Relativista y Radioastronomía

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. The non-linear nature of MHD equations 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 subsidiary quantity can be monitored in part to estimate this. This quantity is the monopole constraint 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. 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 but rather on the system of equations to be solved itself. This is achieved by considering an additional variable suitably coupled to the system through another equation so that, through the evolution, the constraint behavior is kept under control. The advantage of this modification is that now the system is strongly hyperbolic. In the present work we concentrate on formulating constraint preserving boundary conditions for the Newtonian ideal MHD equations. We show how the boundary conditions developed significantly reduce the violations generated at the boundaries at the numerical level and how lessen their influence in the interior of the computational domain by making use of the available freedom in the equations.