Up: Three-dimensional chromospheric magnetic

# Appendix B. The vertical (z) integration

The analysis by Bruma & Cuperman (1996) clearly indicates that the accuracy of the vertical (z) integration in problems of the type considered here, can be significantly improved upon formulating the problem in terms of the value of the function to be integrated (F) at z=0 and its derivatives at (equidistant) grid points above z=0. We here present some final results obtained in the reference indicated above.

Denote by the coordinates of a set of equidistant grid points along the z-axis and by ..... ..... the values of the function F at the corresponding grid points. Let and ; then, about 500 grid points are involved. The first ``moving" set is labeled j=0 and starts at the photosphere, (Notice that here F represents either or

By the aid of the functions ( z=0,1,..,n) it is possible to calculate their first order derivatives, ; then, in the resulting system of relationships, we eliminate the functions ..... as well as the derivative to obtain a relation of the form

Thus, at a grid point we have a ten-grid-point-formula for the function F in terms of the F-value at and its derivatives at all other points. In particular, starting at the photosphere where the above symbolic formula reads

The explicit relation (20 (click here) ) is obtained by the aid of the REDUCE-package and it reads

At this point the following remark is in order: at grid points , some simpler integration formulas can be used, namely:

As proposed and implemented by Bruma & Cuperman (1996), utilization of an appropriate iterative procedure leads to improvement of the zero-order results obtained at very low levels

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