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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 ( h ) = X j = u j e ijh . Verify the following formulas. a) u j = 1 2 Z  b u ( h ) e ijh d ( h ) , b) X j =  u j  2 = 1 2 Z   b u ( h )  2 d ( h ) 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 ( h ) and show that 0 ( h ) 1 for all h . Plot ( h ) for 0 h and = 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
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