Chapter 5
Numerical Methods in Heat Conduction
Special Topic: Controlling the Numerical Error
596C
The results obtained using a numerical method differ from the exact results obtained analytically
because the results obtained by a numerical method are approximate.
The difference between a
numerical solution and the exact solution
(the error) is primarily due to two sources: The
discretization
error
(also called the
truncation
or
formulation
error) which is caused by the approximations used in the
formulation of the numerical method, and the
roundoff error
which is caused by the computers'
representing a number by using a limited number of significant digits and continuously rounding (or
chopping) off the digits it cannot retain.
597C
The
discretization error
(also called the
truncation
or
formulation
error) is due to replacing the
derivatives by differences in each step, or replacing the actual temperature distribution between two
adjacent nodes by a straight line segment. The difference between the two solutions at each time step is
called the
local discretization error
. The total discretization error at any step is called the
global
or
accumulated discretization error
. The local and global discretization errors are identical for the first time
step.
598C
Yes, the global (accumulated) discretization error be less than the local error during a step. The
global discretization error usually increases with increasing number of steps,
but the opposite may occur
when the solution function changes direction frequently, giving rise to local discretization errors of
opposite signs which tend to cancel each other.
599C
The Taylor series expansion of the temperature at a specified nodal point
m
about time
t
i
is
+
∆
+
∆
+
=
∆
+
2
2
2
)
,
(
2
1
)
,
(
)
,
(
)
,
(
t
t
x
T
t
t
t
x
T
t
t
x
T
t
t
x
T
i
m
i
m
i
m
i
m
∂
∂
∂
∂
The finite difference formulation of the time derivative at the same nodal point is expressed as
∂
∂
T x
t
t
T x
t
t
T x
t
t
T
T
t
m
i
m
i
m
i
m
i
m
i
(
,
)
(
,
)
(
,
)
2245
+

=

+
∆
∆
∆
1
or
T x
t
t
T x
t
t
T x
t
t
m
i
m
i
m
i
(
,
)
(
,
)
(
,
)
+
2245
+
∆
∆
∂
∂
which resembles the Taylor series expansion terminated after the first two terms.
598
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Chapter 5
Numerical Methods in Heat Conduction
5100C
The Taylor series expansion of the temperature at a specified nodal point
m
about time
t
i
is
T x
t
t
T x
t
t
T x
t
t
t
T x
t
t
m
i
m
i
m
i
m
i
(
,
)
(
,
)
(
,
)
(
,
)
+
=
+
+
+
∆
∆
∆
∂
∂
∂
∂
1
2
2
2
2
The finite difference formulation of the time derivative at the same nodal point is expressed as
∂
∂
T x
t
t
T x
t
t
T x
t
t
T
T
t
m
i
m
i
m
i
m
i
m
i
(
,
)
(
,
)
(
,
)
2245
+

=

+
∆
∆
∆
1
or
T x
t
t
T x
t
t
T x
t
t
m
i
m
i
m
i
(
,
)
(
,
)
(
,
)
+
2245
+
∆
∆
∂
∂
which resembles the Taylor series expansion terminated after the first two terms. Therefore, the 3rd and
following terms in the Taylor series expansion represent the error involved in the finite difference
approximation. For a sufficiently small time step, these terms decay rapidly as the order of derivative
increases, and their contributions become smaller and smaller. The first term neglected in the Taylor
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 Spring '11
 ENgel
 Numerical Analysis, Heat, Heat Transfer, finite difference

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