NATURAL DISCRETIZATIONS FOR THE DIVERGENCE, GRADIENT, AND CURL ON LOGICALLY RECTANGULAR GRIDS

This is the first in series of papers creating a discrete analog of ve ctor analysis on logically rectangular, nonorthogonal, nonsmooth grids . We introduce notations for 2-D logically rectangular grids, describe both cell-valued and nodal discretizations for scalar functions, and construct the natural discretizations of vector fields, using the vect or components normal and tangential to the cell boundaries. We then de fine natural discrete analogs of the divergence, gradient, and curl op erators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates, such as length of elemen ts of coordinate lines, areas of elements of coordinate surfaces, and elementary volumes. We introduce the discrete volume integral of scala r functions, the discrete surface integral, and a discrete analog of t he line integral and prove discrete versions of the main theorems rela ting these objects. These theorems include the following: the discrete analog of relationship div (A) over right arrow = 0 if and only if (A ) over right arrow = curl (B) over right arrow; curl (A) over right ar row = 0 if and only if (A) over right arrow = grad phi; if (A) over ri ght arrow = grad phi, then the line integral does not depend on path; and if the line integral of a vector function is equal to zero for any closed path, then this vector is the gradient of a scalar function. L ast, we define the discrete operators DIV, GRAD, and CURL in terms of primitive differencing operators (based on forward and backward differ ences) and primitive metric operators (related to multiplications of d iscrete functions by length of edges, areas of surfaces, and volumes o f 3-D cells). These formulations elucidate the structure of the discre te operators and are useful when investigating the relationships betwe en operators and their adjoints.