Geography Reference
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13.5. Suppose that the streamfunction ψ is given by a single sinusoidal wave
ψ(x)
Asin(kx). Find an expression for the error of the finite difference
approximation
=
2 ψ
∂x 2
ψ m + 1
m +
ψ m 1
δx 2
for kδx
=
π/8,π/4,π/2, and π . Here x
=
mδx with m
=
0, 1, 2,....
13.6. Using the method given in Section 13.3.3, evaluate the computational
stability of the following two finite difference approximations to the one-
dimensional advection equation:
σ
ζ m,s ζ m 1,s
(a) ζ m,s + 1 ζ m,s =−
σ
ζ m + 1,s ζ m,s
(b) ζ m,s + 1 ζ m,s =−
where σ
cδt/δx > 0. [The schemes labeled (a) and (b) are referred to as
upstream and downstream differencing, respectively.] Show that scheme
(a) damps the advected field and compute the fractional damping rate per
time step for σ
=
=
=
0.25 and kδx
π/8 for a field with the initial form
=
exp(ikx).
13.7. Using a staggered horizontal grid analogous to that shown in Fig. 13.5
(but for an equatorial β-plane geometry), express the linearized shallow
water equations (11.29)-(11.31) in finite difference form using the finite
differencing and averaging operator notation introduced in Section 13.6.1.
13.8. Verify the equality
ζ
1
exp
2iθ p
i tan θ p
=
1
+
i tan θ p
given in (13.22).
13.9. Compute the ratio of the numerical phase speed to the true phase speed,
c /c, for the implicit differencing scheme of (13.19) for p
=
π, π/2,π/4,
π/8, and π/16. Let σ
=
0.75 and σ
=
1.25. Compare your results to
those of Table 13.1.
13.10. Using the technique of Section 13.3.1, show that the following four-point
difference formula for the first derivative is of fourth-order accuracy:
ψ (x 0 +
4
3
δx)
ψ (x 0
δx)
ψ (x 0 )
2δx
ψ (x 0 +
1
3
2δx)
ψ (x 0
2δx)
(13.72)
4δx
 
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