Biomedical Engineering Reference
In-Depth Information
Fig. 7.2
A representation of a one-dimensional and two-dimensional uniformly distributed cartesian
grid for the finite difference method (full symbols denote boundary nodes and open symbols denote
computational nodes)
7.2.1
Finite Difference Method
In the finite difference (FD) method, the partial derivatives are approximated by
FD equations at each grid nodal point through the Taylor series expansions. These
derivatives, replaced by FD approximations, yield an algebraic equation for the flow
solution at each grid point. In principle, FD can be applied to any type of grid system.
However, the method is more commonly applied to structured grids since it requires
a mesh having a high degree of regularity. The grid spacing between the nodal points
need not be uniform, but there are limits on the amount of grid stretching or distortion
that can be imposed, to maintain accuracy. Additionally the grid is usually taken to
be locally structured, which means that each grid node may be considered the origin
of a local coordinate system, whose axes coincide with the grid lines. These two grid
lines also imply that they do not intersect elsewhere and that any pair of grid lines
belonging to the different families intersects only once at the grid point. In three
dimensions, three grid lines intersect at each node; none of these lines intersect each
other at any other grid nodal point.
Figure
7.2
illustrates a one-dimensional uniformly distributed Cartesian grid com-
monly used in the FD method. Within this grid system, each node is uniquely
identified by an index, which represents nodal points on the grid lines (
i
). In two-
dimensions the indices are (
i
,
j
) and in three-dimensions (
i
,
j
,
k
). The neighbouring
nodes are defined by increasing or reducing one of the indices by unity. The spacing
of the grid points in the
x
direction are assumed to be uniform and given by
x
respec-
tively. The spacing of
x
need not necessarily be uniform and it could have easily
been dealt with using totally unequal spacing in both directions. As is frequently
handled in CFD, the numerical calculations can be performed in a transformed com-
putational space that has uniform spacing in the transformed independent variables
but still corresponds to a non-uniform spacing in the physical space.
Before we introduce the discretisation through the Taylor series expansion for the
FD equations, let us consider a derivative and its definition from first principles as
f
(
a
+
h
)
−
f
(
a
)
f
(
a
)
=
lim
h
→
0
(7.1)
h
then its approximation is simply
f
(
a
+
h
)
−
f
(
a
)
f
(
a
)
≈
(7.2)
h
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