146
3. Beyondthe One-Way WaveEquation
If 0 s JL :::: 1, then as demonstrated in Section 2.2.2, IjhI s 1 and
IAkl s 1 + Irlßt ,
which satisfies the stability criterion (3.86). The finite -difference approximation
(3.84) to the advection-source equation is therefore stable whenever the associated approximation to the pure advection problem is stable.
Clearly, the methodology used in the preceding stability analysis can be generalized to a wider dass of problems. Let L (1/1) be a linear operator involving
partial derivatives of 1/1 and consider the family of partial differential equations of
-
a1/l
at + L(1/I) + r1/l = O.
the form
As demonstrated by Strang (1964), the range of M for which any explicit twotime-level approximation to the preceding partial differential equation satisfies
the stability condition (3.86) is independent of the value of r. Unfortunately, it is
rather easy to misinterpret this result. The Strang perturbation theorem guarantees only that the value of r has no influence on the ability of consistent finitedifference approximations to converge to the correct solution in the limit of ßt,
ßx -+ O. The value of r does affect the boundedness of numerical solutions
computed with finite values of l:i.t and Sx ,
The solution to the advection-sink problem (3.83) is bounded whenever r > 0,
and in such circumstances the numerical approximation obtained using finite Sx
and ßt should satisfy the more strict stability condition IAkI :::: 1. The conditions on r l:i.t required to guarantee a nongrowing solution may be determined as
folIows. Using (3.85),
lAd = (I - Äi - 2JL(I - JL - A)(1 - coskl:i.x) .
Now consider two cases. First, if JL(I - JL - A) ::: 0, the largest amplification
factor occurs when k = 0 and
IAol = (1 - A)2.
In this case all jAkI are less than unity when (l - A)2 :::: 1 or 0 :::: A :::: 2. This
inequality is always satisfied, since r > 0, e > 0, and by assumption JL(1 - JL -
A) ::: O. All the stability restrictions on M arise therefore from the the second
case, for which JL(l - JL - A) < O. In this case the largest amplification factor
occurs for k l:i.x = n and
IA Jf ß x l 2 = (I - A)2 - 4JL(l - JL - A) = (1 - A - 2JL)2 .
Thus alllAkl are less than unity when (l - A - 2JL)2 :::: 1 or 0:::: A+ 2JL :::: 2, or
equivalently,
0 < -
eßt (
rtu)
1+-- <1.
- l:i.x
2 e -
The last expression shows that the value of r ceases to restriet the maximum stable
time step as ßx -+ O. This is consistent with the implication of the Strang per-
3. Beyondthe One-Way WaveEquation
If 0 s JL :::: 1, then as demonstrated in Section 2.2.2, IjhI s 1 and
IAkl s 1 + Irlßt ,
which satisfies the stability criterion (3.86). The finite -difference approximation
(3.84) to the advection-source equation is therefore stable whenever the associated approximation to the pure advection problem is stable.
Clearly, the methodology used in the preceding stability analysis can be generalized to a wider dass of problems. Let L (1/1) be a linear operator involving
partial derivatives of 1/1 and consider the family of partial differential equations of
-
a1/l
at + L(1/I) + r1/l = O.
the form
As demonstrated by Strang (1964), the range of M for which any explicit twotime-level approximation to the preceding partial differential equation satisfies
the stability condition (3.86) is independent of the value of r. Unfortunately, it is
rather easy to misinterpret this result. The Strang perturbation theorem guarantees only that the value of r has no influence on the ability of consistent finitedifference approximations to converge to the correct solution in the limit of ßt,
ßx -+ O. The value of r does affect the boundedness of numerical solutions
computed with finite values of l:i.t and Sx ,
The solution to the advection-sink problem (3.83) is bounded whenever r > 0,
and in such circumstances the numerical approximation obtained using finite Sx
and ßt should satisfy the more strict stability condition IAkI :::: 1. The conditions on r l:i.t required to guarantee a nongrowing solution may be determined as
folIows. Using (3.85),
lAd = (I - Äi - 2JL(I - JL - A)(1 - coskl:i.x) .
Now consider two cases. First, if JL(I - JL - A) ::: 0, the largest amplification
factor occurs when k = 0 and
IAol = (1 - A)2.
In this case all jAkI are less than unity when (l - A)2 :::: 1 or 0 :::: A :::: 2. This
inequality is always satisfied, since r > 0, e > 0, and by assumption JL(1 - JL -
A) ::: O. All the stability restrictions on M arise therefore from the the second
case, for which JL(l - JL - A) < O. In this case the largest amplification factor
occurs for k l:i.x = n and
IA Jf ß x l 2 = (I - A)2 - 4JL(l - JL - A) = (1 - A - 2JL)2 .
Thus alllAkl are less than unity when (l - A - 2JL)2 :::: 1 or 0:::: A+ 2JL :::: 2, or
equivalently,
0 < -
eßt (
rtu)
1+-- <1.
- l:i.x
2 e -
The last expression shows that the value of r ceases to restriet the maximum stable
time step as ßx -+ O. This is consistent with the implication of the Strang per-