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where
u
denotes the solution of our model problem (
8.1
)-(
8.3
). In particular, this
means that
j
u
.x; t /
j
max
x
j
f.x/
j
for all x
2
.0; 1/ and t
0:
Let us see if we can prove a similar property for the approximations generated by
the explicit scheme. To this end, assume that t and x satisfy
t
x
2
1
2
:
˛
D
Note that, if this condition holds, then
1
2˛
0;
(8.85)
and this turns out to be the key point in the present analysis.
To simplify the notation, we introduce the symbol
u
`
for the maximum of the
absolute value of the discrete approximation at time step t
`
,thatis,
u
`
D
max
i
j
u
i
j
for `
D
0;:::;m;
and note that
u
0
D
max
i
j
f.x
i
/
j
:
If (8.85) holds, then it follows from (7.91) and the triangle inequality that
j
u
`
C
1
i
jDj
˛
u
i
1
C
.1
2˛/
u
i
C
˛
u
i
C
1
j
j
˛
u
i
1
jCj
.1
2˛/
u
i
jCj
˛
u
i
C
1
j
D
˛
j
u
i
1
jC
.1
2˛/
j
u
i
jC
˛
j
u
i
C
1
j
˛
u
`
C
.1
2˛/
u
`
C
˛
u
`
D
u
`
(8.86)
for i
D
2;:::;n
1. Moreover, we have
u
`
C
1
1
D
u
`
C
1
n
D
0;
and consequently, since (
8.86
) is valid for i
D
2;:::;n
1,
j
u
`
C
1
i
j
u
`
;
max
i
or
u
`
C
1
u
`
:
