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Unformatted text preview: 1 Atms 502 CS 505 CSE 566 Numerical Fluid Dynamics Thu., 14 September 2006 9/29/06 Atms 502  Fal 2006  Jewett 2 Our original equation: Our expression for u tt was: Substituting, we get: • This is the modified equation • It is what is actually solved by the F.D. method • U xx term: implicit artificial viscosity The modifed equation (2) u tt = c 2 u xx + " t # u ttt 2 + c 2 u ttx + ... $ % & ’ ( ) + " x c 2 u xxt # c 2 2 u xxx + ... $ % & ’ ( ) (4) u t + cu x = " # t 2 u tt + c # x 2 u xx + higher order terms (1) u t + cu x = c " x 2 1 # $ ( ) u xx + (..)( " x ) 2 u xxx + higher order terms 9/29/06 Atms 502  Fal 2006  Jewett 3 The modifed equation (4) Modified equation for the upstream method: u t + cu x = c " x 2 1 # $ ( ) u xx + (..)( " x ) 2 u xxx + higher order terms L=2 ∆ x Anderson et al., chapter 4 L → ʿ No amplitude error for ν =1 (it matches the unit circle)....
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 Spring '09
 JEWETT
 Velocity, Characteristic polynomial, Complex number, group velocity, Jewett

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