11—Numerical Analysis 289Looking more closely though, the object in brackets in Eq. (11.70) has magnitude
r=
[
1 +
c^2 (∆t)^2
(∆x)^2
sin^2 k∆x
] 1 / 2
> 1 (11.71)
so the magnitude of the solution grows exponentially. This instability can be pictured as a kind of
negative dissipation. This growth is reduced by requiringkc∆t 1.
Given a finite fixed time interval, is it possible to get there with arbitrary accuracy by making∆t
small enough? Withnsteps=t/∆t,rnis
r=
[
1 +
c^2 (∆t)^2
(∆x)^2
sin^2 k∆x
]t/2∆t= [1 +α]β
=
[
[1 +α]^1 /α
]αβ≈eαβ
= exp[
c^2 t∆t
2(∆x)^2
sin^2 k∆x
]
,
so by shrinking∆tsufficiently, this is arbitrarily close to one.
There are several methods to avoid some of these difficulties. One is the Lax-Friedrichs method:u(t+ ∆t,x) =
1
2
[
u(t,x+ ∆x) +u(t,x−∆x)
]
−
c∆t
2∆x
[
u(t,x+ ∆x)−u(t,x−∆x)
]
(11.72)
By appropriate choice of∆tand∆x, this will haver≤ 1 , causing a dissipation of the wave. Another
scheme is the Lax-Wendroff method.
u(t+ ∆t,x) =u(t,x)−
c∆t
2∆x
[
u(t,x+ ∆x)−u(t,x−∆x)
]
+
c^2 (∆t)^2
2(∆x)^2
[
u(t,x+ ∆x)− 2 u(t,x) +u(t,x−∆x)
]
(11.73)
This keeps one more term in the power series expansion.
Exercises1 Use the four-point interpolation formula Eq. (11.3) to estimatee^3 /^4 from the values ofexat 0 , 1 / 2 ,
1 , 3 / 2. From the known value of the number, compute the relative error.
2 Find a root of the equationcosx=x. Start with a graph of course.
3 Find the values ofαand ofxfor whichex=αxhas a single root.
4 Find the roots ofex=αxwhenαis twice the value found in the preceding exercise. (and where is
your graph?)
5 Use (a) midpoint formula and (b) Simpson’s rule, with two intervals in each case, to evaluate
4
∫ 1
0 dx^1 /(1 +x
(^2) ).