Divergence-free Condition Of Magnetic Field Research Articles

On the impact of the numerical method on magnetic reconnection and particle acceleration – I. The MHD case

ABSTRACT We present 2D magnetohydrodynamics numerical simulations of tearing-unstable current sheets coupled to a population of non-thermal test particles, in order to address the problem of numerical convergence with respect to grid resolution, numerical method, and physical resistivity. Numerical simulations are performed with the pluto code for astrophysical fluid dynamics through different combinations of Riemann solvers, reconstruction methods, and grid resolutions at various Lundquist numbers. The constrained transport method is employed to control the divergence-free condition of magnetic field. Our results indicate that the reconnection rate of the background tearing-unstable plasma converges only for finite values of the Lundquist number and for sufficiently large grid resolutions. In general, it is found that (for a second-order scheme) the minimum threshold for numerical convergence during the linear phases requires the number of computational zones covering the initial current sheet width to scale roughly as $\sim \sqrt{\bar{S}}$, where $\bar{S}$ is the Lundquist number defined on the current sheet width. On the other hand, the process of particle acceleration is found to be nearly independent of the underlying numerical details inasmuch as the system becomes tearing-unstable and enters in its non-linear stages. In the limit of large $\bar{S}$, the ensuing power-law index quickly converge to p ≈ 1.7, consistently with the fast reconnection regime.

Systematic construction of upwind constrained transport schemes for MHD

The constrained transport (CT) method reflects the state of the art numerical technique for preserving the divergence-free condition of magnetic field to machine accuracy in multi-dimensional MHD simulations performed with Godunov-type, or upwind, conservative codes. The evolution of the different magnetic field components, located at zone interfaces using a staggered representation, is achieved by calculating the electric field components at cell edges, in a way that has to be consistent with the Riemann solver used for the update of cell-centered fluid quantities at interfaces. Albeit several approaches have been undertaken, the purpose of this work is, on the one hand, to compare existing methods in terms of robustness and accuracy and, on the other, to extend the upwind constrained transport (UCT) method by Londrillo & Del Zanna (2004) [22] and Del Zanna et al. (2007) [23] for the systematic construction of new averaging schemes. In particular, we propose a general formula for the upwind fluxes of the induction equation which simply involves the information available from the base Riemann solver employed for the fluid part, provided it does not require full spectral decomposition, and 1D reconstructions of velocity and magnetic field components from nearby intercell faces to cell edges. Our results are presented here in the context of second-order schemes for classical MHD, but they can be easily generalized to higher than second order schemes, either based on finite volumes or finite differences, and to other physical systems retaining the same structure of the equations, such as that of relativistic or general relativistic MHD.

AN EXPLICIT SCHEME FOR INCORPORATING AMBIPOLAR DIFFUSION IN A MAGNETOHYDRODYNAMICS CODE

We describe a method for incorporating ambipolar diffusion in the strong coupling approximation into a multidimensional magnetohydrodynamic (MHD) code based on the total variation diminishing scheme. Contributions from ambipolar diffusion terms are included by explicit finite-difference operators in a fully unsplit way, maintaining second-order accuracy. The divergence-free condition of magnetic fields is exactly ensured at all times by a flux-interpolated constrained transport scheme. The super time-stepping method is used to accelerate the time step in high-resolution calculations and/or in strong ambipolar diffusion. We perform two test problems, the steady-state oblique C-type shocks and the decay of Alfven waves, confirming the accuracy and robustness of our numerical approach. Results from the simulations of the compressible MHD turbulence with ambipolar diffusion show the flexibility of our method as well as its ability to follow complex MHD flows in the presence of ambipolar diffusion. These simulations show that the dissipation rate of the MHD turbulence is strongly affected by the strength of ambipolar diffusion.

A Simple Finite Difference Scheme for Multidimensional Magnetohydrodynamical Equations

An approximate MHD Riemann solver, an approach to maintain the divergence-free condition of magnetic field, and a finite difference scheme for multidimensional magnetohydrodynamical (MHD) equations are proposed in this paper. The approximate MHD Riemann solver is based on characteristic formulations. Both the conservation laws for mass, momentum, energy, and magnetic field, and the divergence-free condition of the magnetic field are exactly satisfied in the proposed scheme. The scheme does not involve any Poisson solver and is second-order accurate in both space and time. The correctness and robustness of the scheme are shown through numerical examples. The approach proposed in this paper to maintain the divergence-free condition may be applied to other dimensionally split and unsplit Godunov schemes for MHD flows.

On the Divergence‐free Condition and Conservation Laws in Numerical Simulations for Supersonic Magnetohydrodynamical Flows

An approach to maintain exactly the eight conservation laws and the divergence-free condition of magnetic fields is proposed for numerical simulations of multidimensional magnetohdyrodynamic (MHD) equations. The approach is simple and may be easily applied to both dimensionally split and unsplit Godunov schemes for supersonic MHD flows. The numerical schemes based on the approach are second-order accurate in both space and time if the original Godunov schemes are. As an example of such schemes, a scheme based on the approach and an approximate MHD Riemann solver is presented. The Riemann solver is simple and is used to approximately calculate the time-averaged flux. The correctness, accuracy, and robustness of the scheme are shown through numerical examples. A comparison in numerical solutions between the proposed scheme and a Godunov scheme without the divergence-free constraint implemented is presented.