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CFD lecture 4

# CFD lecture 4 - Computational Fluid Dynamics Lecture 4...

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Computational Fluid Dynamics Lecture 4 Tri-diagonal matrices are very efficient to solve computationally using the Thomas algorithm. Pg 183-184 using forward substitution and backward elimination. 1 1 1 2 2 2 2 3 3 3 1 1 1 i i i m m m m m b c u a b c u a b c a bc a b c a b % % 1 2 3 3 1 1 i i m m m m R R u R u R u R u R = # # # # Consider the linearized advection-diffusion equation. 2 2 u u C v t x + = u x discretized with the backward Euler method, and second order spatial differences. 1 1 1 1 1 1 1 1 1 2 2 2 n n n n i i i i n n i i u u u u C t x u u u v 1 n i x + + + + + + + = − + + + 1 + There are three unknowns. 1 1 1 1 , , n n n i i i u u u + + + Write system in the form [ ][ ] [ ] 1 1 1 1 1 n n n i i i i i i i Au B u C u R A u R + + + + + + = = 1 1 1 1 1 2 2 Rearranging gives 2 1 2 2 n n n i i i C t v t v t C t v t u u u x x x x x + + + + + + + = 2 n i u special conditions are required near boundaries 1

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i.e. when i is undefined. 1, 1 i = Appropriate boundary conditions might be 0 or 0 B B u u x = = These prescribed B.C.’s take precedence over the governing equation and so the equation at the boundary is just 2 1 1 0, or 0 u u u x = = and with the definitions that 1
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CFD lecture 4 - Computational Fluid Dynamics Lecture 4...

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