Natural discretizations for the divergence, gradient, and curl on logically rectangular grids

Abstract

This is the first in series of papers creating a discrete analog of vector 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 vector components normal and tangential to the cell boundaries. We then define natural discrete analogs of the divergence, gradient, and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates, such as length of elements of coordinate lines, areas of elements of coordinate surfaces, and elementary volumes. We introduce the discrete volume integral of scalar functions, the discrete surface integral, and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects. These theorems include the following: the discrete analog of relationship div A → = 0 if and only if A → = curl B → ; curl A → = 0 if and only if A → = grad ϕ ; if A → = grad ϕ , 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. Last, we define the discrete operators DIV, GRAD, and CURL in terms of primitive differencing operators (based on forward and backward differences) and primitive metric operators (related to multiplications of discrete functions by length of edges, areas of surfaces, and volumes of 3-D cells). These formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints.