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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 tex2html_wrap_inline3001 the coordinates of a set of equidistant grid points along the z-axis and by tex2html_wrap_inline3005 tex2html_wrap_inline3007 ..... tex2html_wrap_inline3011 ..... tex2html_wrap_inline3015 the values of the function F at the corresponding grid points. Let tex2html_wrap_inline3019 and tex2html_wrap_inline3021; then, about 500 grid points are involved. The first ``moving" set is labeled j=0 and starts at the photosphere, tex2html_wrap_inline3027 (Notice that here F represents either tex2html_wrap_inline3031 or tex2html_wrap_inline3033

By the aid of the functions tex2html_wrap_inline3035 ( z=0,1,..,n) it is possible to calculate their first order derivatives, tex2html_wrap_inline3039gif; then, in the resulting system of relationships, we eliminate the functions tex2html_wrap_inline3045 ..... tex2html_wrap_inline3049 as well as the derivative tex2html_wrap_inline3051 to obtain a relation of the form
Thus, at a grid point tex2html_wrap_inline3053 we have a ten-grid-point-formula for the function F in terms of the F-value at tex2html_wrap_inline3059 and its derivatives at all other points. In particular, starting at the photosphere where tex2html_wrap_inline3061 tex2html_wrap_inline3063 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 tex2html_wrap_inline3065, 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 tex2html_wrap_inline3067

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