MATH 517: Extra Problems (Do Not Turn-In)
Spring 2016
1. Consider the numerical scheme:
k n
1 n
n
ri+1 + ri−1
+
si+1 − sni−1 ,
2
2h
k
1 n
n
n
=
si+1 + sni−1 +
ri+1
− ri−1
.
2
2h
rin+1 =
sn+1
i
(a) With which system of PDEs is this scheme consistent?
(b) Find conditions on k and h sufficient for this scheme to satisfy the von Neumann
stability condition.
(c) Find the modified equations for this scheme and show how your von Neumann
stability analysis relates to them.
n , and
2. The Beam-Warming method is a one-sided method using the points Ujn , Uj−1
n+1
n
Uj−2 to update Uj :
a2 k 2
ak
n
n
n
n
Ujn+1 = Ujn −
3Ujn − 4Uj−1
+ Uj−2
+
Ujn − 2Uj−1
+ Uj−2
.
2
2h
2h
(a) Show that this method is second order accurate.
(b) Determine the stability restriction on k for this method.
3. Let F denote the Fourier transform, so that u, v ∈ L2 (lR) have Fourier transforms
û = F{u} and v̂ = F{v}. Verify the following properties (don’t worry about rigorously justifying operations on integrals):
F{u + v}(k) = û(k) + v̂(k);
(a) Linearity.
F{cu}(k) = cû(k).
(b) Translation.
If x0 ∈ lR, then F{u(x + x0 )}(k) = eikx0 û(k).
(c) Modulation.
If k0 ∈ lR, then F{eik0 x u(x)}(k) = û(k − k0 ).
(d) Dilation.
If c ∈ lR with c 6= 0, then F{u(cx)}(k) = û(k/c)/|c|.
F{ū}(k) = û(−k).
(e) Conjugation.
(f) Differentiation.
(g) Inversion.
If ux ∈ L2 (lR), then F{ux }(k) = ikû(k).
F −1 {u}(k) = (2π)−1 û(−k).
4. Differentiation of u(x) = eikx multiplies it by g∞ (k) = ik. Determine the analogous
functions g2 (k) and g4 (k) corresponding to the second and fourth-order finite difference methods. Make a plot of g2 (k), g4 (k), and g∞ (k) versus k. Where in the plot
do we see the order of accuracy of the finite difference formulas?
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