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Appendix A. Horizontal (x,y) derivatives

Following the computational algorithm developed by Bruma & Cuperman (1996) for the calculation of horizontal derivatives in problems of the type studied here, we use a ``flexible" fourteen-grid-point formulation designed to achieve optimal accuracy.

1. Denote by gif tex2html_wrap_inline2939 the coordinates of a set of equidistant grid points along the x-axis and by tex2html_wrap_inline2943 tex2html_wrap_inline2945 .., tex2html_wrap_inline2949 ,.. tex2html_wrap_inline2953 the values of the function F at the corresponding grid points; here F stands for either tex2html_wrap_inline2959 or tex2html_wrap_inline2961. Also, use the notation tex2html_wrap_inline2963. tex2html_wrap_inline2965, where tex2html_wrap_inline2967 represents the equidistant grid size.

2. Concerning the meaning of subscripts used: (a) the subscripts tex2html_wrap_inline2969 define a ``moving" set of fifteen grid-point numbers; (b) the subscripts tex2html_wrap_inline2971 indicate the order of the moving-set along the x-axis, starting at tex2html_wrap_inline2975 (where i = 1 ) and ending at tex2html_wrap_inline2979

3. At a point tex2html_wrap_inline2981, the optimal value of the derivative tex2html_wrap_inline2983 is obtained through a systematic investigation leading to the minimization of the relative error involved. This procedure leads to one of the following two possibilities: (a) use of a symmetric fourteen-point formula, with seven points on each side of the point tex2html_wrap_inline2985 or, (b) use of a non-symmetric variable-number-of-terms-formula tex2html_wrap_inline2987 at the left or at the right of tex2html_wrap_inline2989.

For illustration, in the case (a), one has tex2html_wrap_inline2991
For the case (b)


A complete description on the general algorithm, including the criteria for the selection of the formulas for the horizontal derivations is given in Bruma & Cuperman (1996).

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