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Generating a 2-D coordinate system through differential methods requires
solving for coordinates within domain *X*^{2}.
Incorporating *l* monitor functions for grid regularization expands
the dimensionality of the mapping to,
| |
(9) |

Coordinate system is related to an underlying Cartesian
grid, which is chosen to be a unit square defined by . Transformation is assumed to be
piece-wise smooth and known on the boundary of such that: .
Within this framework, the 2-D gridding equations become,
| |
(10) |

where Laplacian operator and metric tensor *g*_{ij}
are written explicitly Liseikin (2004),
| |
(11) |

| (12) |

One convenient way to solve the set of elliptical
equations 10 is by transforming them to a set of
parabolic equations (i.e. include time-dependence) that have a common
steady-state solution. Thus, equations 10 are
reformulated to include time-dependence - -
leading to the six governing equations,

| |
(13) |

| (14) |

| (15) |

| (16) |

| (17) |

| (18) |

Solutions and satisfying
equations 13-18 will converge to the
solutions of equations 10 as . Hence, the answer to within tolerance factor
occurs at some *T*_{n}. Details of an iterative scheme and an
algorithm for computation to solve equations 13 are
provided in Appendix A.

** Next:** Numerical Examples
** Up:** Shragge: Differential gridding methods
** Previous:** Regularization through Monitor Functions
Stanford Exploration Project

4/5/2006