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4.5 Numerical Stability by Energy Arguments
Finally, we use the inequality
13
(a + b )
ab ≤
2
2
to obtain
n
n

m
m
) + ((v
v
+ v
(v
2
m+1
2
) ≤
j
2
j−1
j+1
j+1
j−1
j=1
j=1
n

) +(v ) ).
((v
≤
m+1 2
j
j=1 1 ((v j j m+1 2 2 1 m 2 m ) +(v m ) )) 147
Collecting these three inequalities, it follows from (4.47) that
n
m
E m+1 − E m ≤ r(E m+1 − E ) − r(1 − 2r)∆x (v m − v )
m 2
j
j+1
j=1
m
≤ r(E m+1 − E ),
where we have used the stability assumption (4.44). Hence,
m
(1 − r)(E m+1 − E ) ≤ 0,
and by (4.44) this implies the desired inequality (4.46).
We summarize the result of the discussion above:
m
Theorem 4.1 Let {v } be a solution of the finite difference scheme (4.42)–
j
m
(4.43) and let the corresponding energy {E } be given by (4.45). If the
m
stability condition (4.25) holds, then {E } is nonincreasing with respect to
m.
Hence, we have seen that the stability condition (4.25), or (4.44), implies
that the explicit difference scheme admits an estimate which is similar to
the estimate (3.60) for the continuous problem. As for the continuous prob-
lem, this can be used to estimate the difference between two solutions, with
different initial data. This follows since the difference of two solutions of
the finite difference scheme is a new finite difference solution. We therefore
obtain
Corollary 4.1 Assume that the stability condition (4.25) holds and let
m
m
{v } and {w } be two solutions of the finite difference scheme (4.42)–
j j
(4.43). Then, for all m ≥ 0,
n
0
m
∆x (v − w ) ≤ ∆x (v − w ) .
0
m 2
0 2
j j j j
j=1 j=1
13 See Exercise 4.24.