Biomedical Engineering Reference
In-Depth Information
For this problem, we could have obtained the solution using the Gaussian elimina-
tion instead of the Thomas (the TDMA matrix) algorithm. For such a small matrix,
the additional arithmetic operations for Gaussian elimination may not be as signifi-
cant compared to the Thomas algorithm. Nevertheless, this is not true when a larger
number of grid or nodal points are used. This is because of the additional and more
cumbersome numerical computations (multiplication and divisions) that have to be
performed on the matrix entries. The algorithm degenerates and becomes inefficient
once the order of the matrix becomes higher (> 10).
The Jacobi Method
To illustrate the
Jacobi method
, the resulting set of algebraic
equations as previously used is rewritten
3000
T
- 1000
T
+×+×=
0
T
0
T
203000
1
2
3
4
-1000
T
+
2000
T
- 1000
T
+× =
0
T
3000
1
2
3
4
0
×
T
- 1000
T
+
2000
T
- 1000
T
=
3000
1
2
3
4
0
×+×
T
0
T
- 1000
T
+ =
3000
T
803000
1
2
3
4
The above set of equations can be reorganized so that the required variable is on the
left hand side of the equation.
TT
1
=
(/ )( /)
2
3233
+
0
TT
=
(
/2) (
+
T
/2)
+
3/2
2
1
3
TT
=
(
/2)
+
(
T
/2)
+
3/2
3
2
4
TT
4
=
(/)( / )
3
3833
+
00
()
0
()
0
()
0
()
0
By employing initial guesses:
TTTT
1
=
=
100
=
, the nodal tempera-
=
2
3
4
tures for the first iteration are determined as:
T
1
1
13233 11
()
=
(
00
/ )( /) .
+
0
=
0 000
T
2
1
121232 115
()
=
(
00
/ )( /) /
+ +=
00
0 0
.
T
3
1
121232 115
()
=
(
00
/ )( /) /
+ +=
00
0 0
.
T
4
1
1383331
()
=
(
00
/ )( /) .
+
0
=
0 000
The above first iteration values of
T
1
1
()
=
,
T
2
1
()
,
T
3
1
()
101
=
101 5
.
=
101 5
.
and
T
4
1
()
=
2
3
317
are substituted back into the system of equations; the second iteration
yields
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