Moreover, usually the underlying simulation gives only and
, so
must be numerically
differentiated to obtain
. As discussed by Thornburg [156
, Section 6.1], it is somewhat more efficient to
combine the numerical differentiation and interpolation operations, essentially doing the differentiation inside the
interpolator27.
Thornburg [156, Section 6.1] argues that for an elliptic-PDE algorithm (Section 8.5), for best
convergence of the nonlinear elliptic solver the interpolated geometry variables should be smooth
(differentiable) functions of the trial horizon surface position. He argues that that the usual
Lagrange polynomial interpolation does not suffice here (in some cases his Newton’s-method
iteration failed to converge) because this interpolation gives results which are only piecewise
differentiable28.
He uses Hermite polynomial interpolation to avoid this problem. Cook and Abrahams [51
], and
Pfeiffer et al. [124
] use bicubic spline interpolation; most other researchers either do not describe their
interpolation scheme or use Lagrange polynomial interpolation.
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