By Taylor's expansion, we have
JR31 = O(h2). (4.4)
Combining (4.2), (4.3), and (4.4), we see that when the solution to (3.10) is
substituted in (3.12), we obtain
RI + R2 + R3 = O(At)m + 0(h2),
where m = 3 if p = 1/2 and m = 2 if p 1/2. m
Now if we add the force term J back to Equation (3.10), then the correspond-
ing finite difference scheme is
A ijk = kAhI,'j,k -2(1 -2 )-9zij,k + I-Uijk]
1 e-(t) (V j)n+ (4.5)
7k E
Lemma 4.2.3 If we use the finite difference scheme (4.5) to discretize Equation
(3.7), then it has the truncation error
O(At)m + O(h2),
where m = 3 if p = 1/2 and m = 2 if p 0 1/2.
Proof: We do not include the force term J in Lemma 4.2.2. Now we only need
to focus on the error caused by estimating the J term when the finite difference
scheme (4.5) is applied to Equation (3.7). In the scheme (4.5). we approximate the
V J term in (3.7) by the V J term in (4.5). By Lemma 4.2.1, we have
e -t+ ds e)f -t+)ds = 0(At)3.