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We present a constrained formulation of Dedner et al's hyperbolic/parabolicdivergence cleaning scheme for enforcing the \nabla\dot B = 0 constraint inSmoothed Particle Magnetohydrodynamics (SPMHD) simulations. The constraint weimpose is that energy removed must either be conserved or dissipated, such thatthe scheme is guaranteed to decrease the overall magnetic energy. This is shownto require use of conjugate numerical operators for evaluating \nabla\dot B and\nabla{\psi} in the SPMHD cleaning equations. The resulting scheme is shown tobe stable at density jumps and free boundaries, in contrast to an earlierimplementation by Price & Monaghan (2005). Optimal values of the dampingparameter are found to be {\sigma} = 0.2-0.3 in 2D and {\sigma} = 0.8-1.2 in3D. With these parameters, our constrained Hamiltonian formulation is found toprovide an effective means of enforcing the divergence constraint in SPMHD,typically maintaining average values of h |\nabla\dot B| / |B| to 0.1-1%, up toan order of magnitude better than artificial resistivity without the associateddissipation in the physical field. Furthermore, when applied to realistic, 3Dsimulations we find an improvement of up to two orders of magnitude in momentumconservation with a corresponding improvement in numerical stability atessentially zero additional computational expense.

Constrained Hyperbolic Divergence Cleaning for Smoothed Particle Magnetohydrodynamics