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Unformatted text preview: MATH 572
Assignment #7 Numerical Methods for Scientiﬁc Computing II Winter 2005 due : Tuesday, April 19 LF = Lax-Friedrichs , LW = Lax-Wendroﬀ
1. Consider the scalar 1D wave equation vt + cvx = 0 with c > 0.
a) Plot (using Matlab) the ampliﬁcation factor ρ(ξh) in the complex plane for the upwind,
LF, and LW schemes, for λ = 1/4, 1/2, 3/4, 5/4. Make a plot for each value of λ, showing
all three methods on the same plot. Include a dotted line showing the unit circle. How are
the methods similar? diﬀerent?
b) Find the model equation for the LF and LW schemes. Compare these to the model
equation for the upwind scheme (derived in class). Which scheme has the most artiﬁcial
viscosity - upwind, LF, or LW? Which scheme has the least?
c) The numerical wave speed has an expansion of the form c = c(1 + γ1 ξh + γ2 (ξh)2 ) + · · ·)
in the long wave limit ξh → 0. Find γ1 , γ2 for the upwind and LF schemes (LW was done
in class). Compare the size of the phase error in the three schemes.
2. Consider the LF scheme for vt + cvx = 0, where c may be positive or negative.
a) Show that the scheme is stable in the ∞-norm if |c|λ ≤ 1.
b) Show that the scheme converges in the ∞-norm if |c|λ ≤ 1.
3. Problem 3 on hw6 asked you to compute the solution of vt + vx = 0 using the upwind
scheme for two initial data. Repeat now using LF and LW. Are the present results better
or worse than those obtained using the upwind scheme?
4. Consider the equation φt + φxxx = 0 and the forward Euler/centered-diﬀerence scheme,
un+1 = un + kD0 D+ D− un . What condition on the parameters h, k is necessary to ensure
stability of the scheme in the 2-norm?
The ﬁnal exam is on Wednesday, April 27, 4-6pm in room 130 Dennison. The exam will
cover the entire course, but PDEs will be emphasized. You may use two pages of notes
(i.e. 2 × 8.5 in × 11 in). Calculators are not allowed. ...
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