42
2. Basic Finite-DifferenceMethods
Squaring both sides and summing over all j gives
L(rPj+I)2 = L [(I -/-L)2(rP'})2 + 2/-L(1 -/-L)rP'}rP'}-1 + /-L2(rP'}_1)2J .
j
j
(2.18)
Assuming cyclic boundary conditions.I
L(rP'}_1)2 = L(rP'})2,
(2.19)
j
j
and using the Schwarz inequality (which states that for two vectors u and v, [u .
v] s lIullllvll),
(2.20)
If /-L(1 -/-L) 2: 0, all three coefficients in (2.18) are positive, and (2.19) and (2.20)
may be used to construct the inequality
L(rPr
J)2:s
[(1 _/-L)2 + 2/-L(l -/-L) + /-L2J L(rP'})2 = L(rP'})2 , (2.21)
j
j
j
which requires IIrP
n
+ 11I2 :s IIrP°1l2 and implies that the scheme is stable. The
condition used to obtain (2.21),
(2.22)
is therefore a sufficient condition for stability. Under the assumption that /-L > 0,
division of (2.22) by /-L leads to the relation /-L :s I, and the total constraint on
/-L is therefore 0 < /-L :s I. A similar treatment of the case /-L :s 0 leads to the
contradictory requirement that /-L 2: land provides no additional solutions. Thus,
recalling the definition of /-L and noting that /-L = 0 satisfies (2.22), the stability
condition may be written
eßt
0<-<1.
- ßx -
As is typical with most conditionally stable difference schemes, there is a maximum limit on the time step beyond which the scheme is unstable, and the stability
limit becomes more severe as the spatial resolution is increased .
2If more general boundary cond itions are imposed at the edges of the spatial domain, a rigorous stability analysis becomes much more difficult. The determination of stability in the presence of
nonperiodie boundaries is discussed in Section 8.1.6.