A&A Supplement series, Vol. 122, April II 1997, 381-396

Received March 14; accepted July 22, 1996

**S. Cuperman - C. Bruma - D. Heristchi**

*Send offprint request: *S. Cuperman

DASOP, Observatoire de Paris, Section Meudon,
F-92195 Meudon Principal Cedex,
France

The three-dimensional (3D) reconstruction of magnetic configurations
above the photosphere is considered within the framework of the
nonlinear force-free-field (FFF) model. The physical- computational algorithm
proposed and tested incorporates, *for the first time*, the following
basic features:
1) Both *photospheric* vector field,
and *chromospheric* line of sight
field component, data are utilized; this reduces
significantly the degree of ill-posedness characterizing the Cauchy
problem corresponding to the case when only - values
are used as boundary conditions.
2) A high-order, very efficient computational algorithm is developed and
used: *horizontal derivatives* are evaluated by 14 - terms
formulas in 14 different forms, selected such as to provide
optimal computational accuracy; the vertical integration
is achieved by the use of ``moving" 10 - term formulas expressed in terms
of 10 derivatives and the first values (*i*=*x*,*y*,*z*).
3) At neutral points, where inherent computational singularities in
the values of the FFF-function arise, rather than using
smoothing techniques based on four-neighbouring- values averages,
suitable procedures ensuring continuity are developed and used.
The overall result of the incorporation of these novel features is an
improvement by orders of magnitude of the accuracy with which the
chromospheric fields are reconstructed in the case in which one uses
(i) only - values as boundary conditions and
(ii) relative simple computational formulas and smoothing techniques;
at , !
The elimination/minimization of measurement errors as well as the
fitting of the corrected date to FFF-model-states is also discussed.

**keywords:** MHD -- methods: numerical -- Sun: magnetic fields

- 1. Introduction
- 2. General formulation and basic equations
- 3. Computational algorithm
- 4. Implementation of the computation alalgorithm-results
- 5. Discussion
- 6. Summary
- Appendix A.
- Appendix B.
- References

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