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 ( 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 free-space 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 2-norm. c) Use the energy method to show that the scheme is unconditionally stable in the 2-norm....
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