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x 10 −4
1.5
1
0.5
0
−0.5
−1
−1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 7.15 The solid line represents the solution (7.106). The dotted , dash-dotted ,and dashed lines
are the numerical results implied by Algorithm 7.1 in the cases of n
D
D
10 and m
17, with
D
D
D
D
D
D
˛
0:4765 (x
1=9), ˛
0:4402 (x
1=19), and ˛
0:4931 (x
1=59), respectively
eventually the numbers produced do not provide an approximation of the solution
of the diffusion equation. This phenomenon is illustrated in Figs. 7.16 and 7.17 :
- If, in the case of x D 1=59, t is reduced from 10=706 to 10=681, and thus
˛ is increased from ˛ D 0:4931 to ˛ D 0:5112, then oscillations occur in the
numerical approximation.
- A further increase of t , corresponding to ˛ D 0:5157, results in a curve that
does not bear any resemblance with the graph of the solution of the problem.
Above we observed that, if ˛ 0:5, then our numerical method for the diffusion
equation works fine. On the other hand, for discretization parameters x and t
such that ˛>0:5, this method did not provide acceptable results. Is this a special
feature of the problem studied in this example or is this a more general disadvantage
of this scheme? To gain further insight, let us perform more experiments.
Now we turn our attention to the diffusion problem
@t D @ 2 u
@ u
for x 2 .0; 1/; t > 0;
@x 2
u .0; t / D u .1; t / D 0 for t>0;
u .x; 0/ D x.1 x/ D x x 2
for x 2 .0; 1/ :
The exact solution to this problem is derived in the next chapter and listed in (8.84)
on page 392. How does the numerical solution produced by Algorithm 7.1 compare
with the exact solution in this case?
 
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