Lecture 34
Finite Difference Method – Nonlinear
ODE
Heat conduction with radiation
If we again consider the heat in a metal bar of length
L
, but this time consider the effect of radiation
as well as conduction, then the steady state equation has the form
u
xx

d
(
u
4

u
4
b
) =

g
(
x
)
,
(34.1)
where
u
b
is the temperature of the background,
d
incorporates a coefficient of radiation and
g
(
x
) is
the heat source.
If we again replace the continuous problem by its discrete approximation then we get
u
i
+1

2
u
i
+
u
i

1
h
2

d
(
u
4
i

u
4
b
) =

g
i
=

g
(
x
i
)
.
(34.2)
This equation is nonlinear in the unknowns, thus we no longer have a system of linear equations
to solve, but a system of nonlinear equations. One way to solve these equations would be by the
multivariable Newton method. Instead, we introduce another interative method.
Relaxation Method for Nonlinear Finite Differences
We can rewrite equation (34.2) as
u
i
+1

2
u
i
+
u
i

1
=
h
2
d
(
u
4
i

u
4
b
)

h
2
g
i
.
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 Fall '08
 Young,T
 Thermodynamics, Equations, Approximation, Convergence, Elementary algebra, initial guess

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