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 freespace 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 2norm. c) Use the energy method to show that the scheme is unconditionally stable in the 2norm....
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 Fall '08
 CONLON
 Math, Numerical Analysis, Formulas, Euler, heat equation vt

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