Differential Equations Lecture Work Solutions 321

Differential Equations Lecture Work Solutions 321 - u n +1...

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1. Use a von Neumann stability analysis to show for the wave equation that a simple explicit Euler predictor using central diFerencing in space is unstable. The diFerence equation is u n +1 j = u n j c t ax u n j +1 u n j 1 2 ! Substitute a ±ourier mode λ n e ik m j x where k m = L , x = L N ,m =0 , 1 ,...,N we get λ n +1 e ik m j x = λ n e ik m j x ν 2 λ n e ik m ( j +1)∆ x e ik m ( j 1)∆ x or λ =1 ν 2 e ik m x e ik m x | {z } 2 i sin k m x Taking absolute value | λ | = ± Re ( λ ) 2 + Im ( λ ) 2 = ± 1+ ν 2 sin 2 k m x> 1 This is always greater than 1 since the second term under the radical is positive. Therefore the method is unstable. Now show that the same diFerence method is stable when written as the implicit formula
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Unformatted text preview: u n +1 j = u n j c t x u n +1 j +1 u n +1 j 1 2 ! As before = 1 i sin k m x or = 1 1 + i sin k m x = 1 i sin k m x 1 + 2 sin 2 k m x Taking absolute value | | = Re ( ) 2 + Im ( ) 2 = v u u t 1 (1 + 2 sin 2 k m x ) 2 + 2 sin 2 k m x (1 + 2 sin 2 k m x ) 2 so | | = s 1 1 + 2 sin 2 k m x 1 This is always less than or equal 1 since the denominator is larger than numerator. Therefore the method is always stable. 321...
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