math572_hw5

# math572_hw5 - MATH 572 Numerical Methods for Scientific...

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Unformatted text preview: MATH 572 Numerical Methods for Scientific Computing II Winter 2005 Assignment #5 due : Thursday March 24 1. Given u j , j = 0 , ± 1 , . . . , define b u ( ξh ) = ∞ X j =-∞ u j e ijξh . Verify the following formulas. a) u j = 1 2 π Z π- π b u ( ξh ) e- ijξh d ( ξh ) , b) ∞ X j =-∞ | u j | 2 = 1 2 π Z π- π | b u ( ξh ) | 2 d ( ξh ) 2. Consider the difference scheme u n +1 j = u n j + kD + D- u n +1 j for the heat equation v t = v xx , corresponding to backward Euler in time and 2nd order central differencing in space. Assume free-space boundary conditions. a) Find the amplification factor ρ ( ξh ) and show that 0 ≤ ρ ( ξh ) ≤ 1 for all ξh . Plot ρ ( ξh ) for 0 ≤ ξh ≤ π and λ = 1 2 , 1 , 10. b) Use Fourier analysis to show that the scheme is unconditionally stable in the 2-norm. c) Use the energy method to show that the scheme is unconditionally stable in the 2-norm....
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