It is also possible to store and evolve point values of the magnetic vector potential, interpolate this vector potential, and find a value and derivatives of the magnetic field from this interpolation. The projection method, used in finite-difference ( Brackbill & Barnes 1980), and pseudo-spectral methods, constructs an interpolation of the magnetic field and then modifies the point values so that with the given interpolation scheme they produce a divergence-free continuous field. The second class of methods constrains the derivatives of the interpolated field. B≠ 0 contributions to the momentum equation.The Smoothed Particle Hydrodynamics schemes of Price & Monaghan (2004a, b, 2005), Børve, Omang & Trulsen (2001) and Dolag & Stasyszyn (2009) also fall into this class, as the former uses a formulation of the MHD equations which is consistent even in ∇ 1999), and diffusion method ( Dedner et al. Methods of this type include the eight-wave scheme ( Powell 1994 Powell et al. B≠ 0 errors, and then attempt to manage the consequences.Two classes of approaches have been used. However, when the magnetic field is represented by point values, the divergence of the interpolated field is not constrained by the point values, so some extra freedom exists. These volume averages constrain the possible divergence of a vector field interpolating these values, and hence the Constrained Transport method ( Evans & Hawley 1988) can be applied to conserve this divergence throughout the simulation. Finite-volume discretizations store the volume average of the field over some cell. Two classes of discretizations are popular in astrophysical applications, finite-volume and point values. In a numerical method, the vector fields are represented by a discrete set of values. B= 0 means, specifying the manner in which B is represented is essential.Brackbill & Barnes 1980 Balsara & Kim 2004 Price 2010 Dolag & Stasyszyn 2009). This issue has attracted much attention in computational astrophysics (e.g. B= 0 constraint in magnetohydrodynamics (MHD) may lead to numerical instability and unphysical results.Magnetic fields, magnetohydrodynamics (MHD), methods: numerical 1 IntroductionĪs originally laid out by Brackbill & Barnes (1980), failing to obey the ∇
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