Day25-slides - 1 Atms 502 Numerical Fluid Dynamics Thu.,...

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Unformatted text preview: 1 Atms 502 Numerical Fluid Dynamics Thu., Nov. 17, 2006 12/6/06 Atms 502 - Fall 2006 - Jewett 2 Diffusion We considered several approaches. Add up terms, as in computer pgm #5, e.g. u3=u1+(advection)+(diffusion)+.. Compute tendencies (contribution) from each, and add, to get A (n+1) Compute one process separately (e.g. advection), update , and use this result in next process This is known as process splitting 12/6/06 Atms 502 - Fall 2006 - Jewett 3 1-step filter to, e.g., remove 2 x in one pass. Add damping term to PDE " = (1 # S ) " j + S 2 " j + 1 + " j # 1 ( ) Diffusion: from last time d " j dt = # " j + 1 $ 2 " j + " $ 1 ( ) 12/6/06 Atms 502 - Fall 2006 - Jewett 4 Diffusion Amplification factor Given: The exact amplification factor is: And this tells us : smallest waves damped most. " ( x , t ) = " e ikx e # Mk 2 t A e = " ( t + # t ) " ( t ) = e $ Mk 2 # t ; if r = M # t # x 2 , A e = e $ Mk 2 # t = e $ r # x 2 k 2 = e $ r % 2 A e = e " r # 2 2 12/6/06 Atms 502 - Fall 2006 - Jewett 5 Diffusion Stability example - FTCS Amplification factor: Damps 2 x most strongly Stable for 0< (1/2) But >(1/4) has <0 for 2 x: sign flips each step....
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Day25-slides - 1 Atms 502 Numerical Fluid Dynamics Thu.,...

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