CFD lecture 5 - Computational Fluid Dynamics Lecture 5 Time...

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Computational Fluid Dynamics Lecture 5 Time differencing continued A. Three level schemes. B. Modified L-F schemes. C. Higher order methods. Three level schemes () ( ) () 11 1 12 1 2 22 2 consistent schemes if 1 and 1 It will be purely explicit and will be second order accurate if: 1 1 3 2 1 1 2 if 0, Leap Frog nn n n n tF tF t ψ φαφ α φ β αα βα +− →= + + + += =− =+ = nd 2 If 0 - 2 order AB2 Adams Bashforth 0: 0 k k b Ci tx t ψψ = ∂∂ + C k b = / one fourier mode With Fourier Series, e.g, 0 N ik x k k bte = = Oscillation equation: k k b iCkb t = , similar properties to wave equation. L-F on Oscillation equation 2 2 21 nnn ik t AA Ai k t A 1 0 n φφ =+∆ == −∆− = Factor out 1 n 1
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Every step has same amplification factor because this is a linear equation with constant coefficients. A has two roots. () 1 22 2 1 Ai k t kt ± =∆±−∆ In limit of good resolution 0 ∆→ 1 and 1 AA +− →→ The solution has two different modes, A + is the physical mode and A is the computational mode. 2 1 2 1 2 1 2 If 1 then 1 is real and A 1 1 and there is no amplitude error in leap frog scheme. In the case 1 11 1 1 which has k t kt kt k t k t i k tkt ± ± + ∆≤ − ∆ =− ∆ + ∆ = ∆>  =∆±− ∆−  =∆± =∆ +∆  1 2 positive 2 magnitude greater than . 1 since 1 so 1 since 1 is unstable. When 1 similarly 1 and when 1 then constant, and thus each time s ik t ikt k t i k t A A i ++ + ∆+ ∆ − > >> > ∆<− > = ±²³²´ D tep produces a 90 shift in the phase of the oscillation, and the unstable mode grows at a period of 4 t. D Relative Phase errors: 1 1 2 1 tan 1 LF R = −∆
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This note was uploaded on 06/08/2011 for the course EOC 6850 taught by Professor Slinn during the Spring '07 term at University of Florida.

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CFD lecture 5 - Computational Fluid Dynamics Lecture 5 Time...

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