Page 87 - Basic Structured Grid Generation

4

Structured grid

generation – algebraic methods

4.1 Co-ordinate transformations

Boundary-value problems of physics and engineering may usually be expressed in
terms of partial differential equations (the hosted equations) to be solved for certain
field quantities as functions of space and possibly time over some region of space,
subject to certain specified boundary (and possibly initial) conditions. When the equa-
tions are expressed in terms of cartesian co-ordinates, a standard numerical method of
solution is through finite differences, where a uniform rectangular grid of regularly-
arranged points is constructed to cover the physical region of space (more precisely,
its mathematical representation) and the partial derivatives in the equations are approx-
imated in terms of the differences between values of the field quantities at adjacent
points of the grid. The resulting equations may then in principle be solved by algebraic
methods.
The essential ideas of this section may be illustrated in two dimensions, with the
Oxy plane divided into equal rectangles with sides of length h, k parallel to the axes
(Fig. 4.1), and grid points x = ih, y = jk, with i,j taking integer values. Denoting
values of a field variable f (x, y) at the i, j grid-point by f i,j , we typically approximate
first derivatives at this point by either forward differences
∂f 1
(f i+1,j − f i,j ), (4.1)
∂x h
or backward differences
∂f 1
(f i,j − f i−1,j ), (4.2)
∂x h
each with ‘first-order’ accuracy, or, with greater (‘second-order’) accuracy, by central
differences
∂f 1
(f i+1,j − f i−1,j ), (4.3)
∂x 2h
and similarly for ∂f/∂y. For second derivatives we have
2
∂ f 1 (f i+1,j − 2f i,j + f i−1,j ), (4.4)
∂x 2 h 2
2
∂ f 1 (f i,j+1 − 2f i,j + f i,j−1 ), (4.5)
∂y 2 k 2