### 6.1 Upward integration method

This straightforward method was proposed by Nakagawa (1974) and it has been first computationally
implemented by Wu et al. (1985, 1990). The basic idea of this method is to reformulate Equations (2) – (4)
and extrapolate the magnetic field vector into the solar corona. The method is not iterative and
extrapolates the magnetic field directly upward, starting from the bottom layer, where the field is measured.
From one computes the -component of the electric current by Equation (13) and
the corresponding -distribution with Equation (14). Then the - and -components of the electric
current are calculated by Equation (11):
Finally, we get the -derivatives of the magnetic field vector with Equations (3) and (4) as
A numerical integration provides the magnetic field vector at the level . These steps are repeated in
order to integrate the equations upwards in . Naively one would assume to derive finally the 3D
magnetic fields in the corona, which is indeed the idea of this method. The main problem is that this simple
straightforward approach does not work because the method is mathematically ill-posed and the algorithm
is unstable (see, e.g., Cuperman et al., 1990 and Amari et al., 1997 for details). As a result of this
numerical instability one finds an exponential growth of the magnetic field with increasing height. The
reason for this is that the method transports information only from the photosphere upwards. Other
boundary conditions, e.g., at an upper boundary, either at a finite height or at infinity cannot
be taken into account. Several attempts have been made to stabilize the algorithm, e.g., by
smoothing and reformulating the problem with smooth analytic functions (e.g., Cuperman
et al., 1991; Démoulin and Priest, 1992; Song et al., 2006). Smoothing does help somewhat
to diminish the effect of growing modes, because the shortest spatial modes are the fastest
growing ones. To our knowledge the upward integration method has not been compared in
detail with other NLFFF codes and it is therefore hard to evaluate the performance of this
method.