CFD lecture 3 - Computational Fluid Dynamics Lecture 3...

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Computational Fluid Dynamics Lecture 3 Discretization Continued…. A fourth order approximation to f x can be found using Taylor Series. () ( ) ( ) ( ) ( ) ( ) 00 0 0 0 0 0 0 0 0 2 2 0: 22 1 : 44 0 : 88 0 : 16 16 v a fx x b x c d x e x f x abcde abd e abd e 0 +∆ + + −∆ + −∆ = ++++= +−− = ′′ +++ = ′′′ +++ = 4 th order solution: ( ) 0 0 41 32 3 4 x x x x xx +∆ −∆ + ∆ − ∆ ∆∆ Another Example: 1 st derivative near a boundary 1 st grid point 3 rd order one sided Dirichlet B.C.using Taylor Series expansions at grid points 2, 3, and 4 with hx = ∆ . [] 11 2 3 4 23 4 21 4 31 4 1 1 2 1 1 26 4 3 99 3 14 3 9 3 2 fA f B f C f D f h hh f f hf f f h f f hf h f h f h f f hf h f h f h f Bf Bhf B f B f Af Cf C hf C h f C h f h Df D hf D h σ =+ + + =+ + + + + + + + + + + + + + ++ + ++ + 3 9 2 fD h f + 1
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() 1 11 2 1 0 23 9 20 2 49 0 63 2 ABCD f h fBCD f BC D h f BC D hf +++ = ′′ =++  ++ =   ′′′ = _____________________________________________________________________________ 12 1 1111 0 0123 1 19 02 0 22 149 0 0 632 Matrix inversion gives: 11 1.8333 6 18 3.0 6 39 1.5 26 36 which agrees exactly with page 61 of ATP. 1 11 18 6 A B C D A B C D f ff zz     =   =− == =−+ ∂∆ [] 34 92 −+ Consistency: The Finite difference approximation is consistent if the truncation error for the whole equation goes to zero as the mesh size goes to zero. Accuracy is rate that F.D. approximation approaches zero as 0 x .
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CFD lecture 3 - Computational Fluid Dynamics Lecture 3...

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