Biomedical Engineering Reference
In-Depth Information
5.2.5
The Algebraic Equation system
Table 5.3
Correspondence table of indices on a grid
Grid location
Notation
Location
i
,
j
,
k
P
I =
(
k
-
1)
N
j
N
i
+
(
i
-
1)
N
j
+
j
i
-
1,
j
,
k
W
I
-
i
,
j
-
1,
k
l
-
1
S
l
+
1
i
,
j
+ 1,
k
N
l
+
N
j
i
+ 1,
j
,
k
E
i
,
j
,
k
-
1
B
l
-
N
i
N
j
i
,
j
,
k
+ 1
T
l
+
N
i
N
j
A inite-difference approximation provides an algebraic equation
at each grid node; it contains the variable value at that nodes as well
as values at neighboring nodes. If the differential equation is non-
linear, the approximation will contain some non-linear terms. The
numerical solution process will then require linearization. We only
consider the linear case. The methods described are applicable in
the non-linear case as well. For this case, the result of discretization
is a system of linear algebraic equations of the form:
ยค
A
A
Q
(5.60)
PP
G
ll
G
P
l
where
P
denotes the node at which the PDE is approximated and
index
l
runs over the neighbor nodes involved in inite-difference
approximations. The node
P
and its neighbors from the so-called
computational molecule; two examples, which result from second-
and third-order approximations, are shown in Figs. 5.12 and 5.13.
The coeficients
A
l
depend on geometrical quantities luid properties
and, for non-linear equations, the variable values themselves.
Q
P
contains all the terms that do not contain unknown variable values;
it is presumed known.
The numbers of equations and unknowns must be equal, i.e.,
there has to be one equation for each grid node. Thus, we have a large
set of linear algebraic equations, which must be solved numerically.
This system is sparse meaning that each equation contains only a
few unknowns. The system can be written in the matrix notation as
follows:
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