In the following, we review five different approaches for the computation of nonlinear force-free coronal magnetic fields. The aim of all codes is to extrapolate photospheric vector field measurements into the corona, but the way how the measurements are used is different. MHD relaxation and optimization methods prescribe the three components of the magnetic field vector on the bottom boundary. Grad–Rubin methods use the vertical magnetic field and the vertical electric current density (or -distribution) as boundary condition. The upward integration method and the boundary-element method require a combination of both boundary conditions. In the following, we will briefly discuss the main features of these five methods. Grad–Rubin, MHD relaxation, and optimization methods require first the computation of a potential field; then the appropriate boundary conditions are specified and eventually one iterates numerically for a solution of the NLFFF equations. Upward integration and boundary-element methods do not require first the computation of a potential field, but solve the NLFFF equations more directly. Both methods have, however, some shortcomings as explained later. Often one is interested anyway to get also the potential field, e.g., to derive the energy the NLFFF field has in excess of the potential field. A more detailed review on the mathematical and computational implementations and recent code updates can be found in Wiegelmann (2008).

6.1 Upward integration method

6.2 Grad–Rubin method

6.3 MHD relaxation method

6.4 Optimization approach

6.5 Boundary-element methods

6.6 Comparison of methods and the NLFFF consortium

6.7 Application of nonlinear force-free codes

6.2 Grad–Rubin method

6.3 MHD relaxation method

6.4 Optimization approach

6.5 Boundary-element methods

6.6 Comparison of methods and the NLFFF consortium

6.7 Application of nonlinear force-free codes

Living Rev. Solar Phys. 9, (2012), 5
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