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. 11 1 222 2 333 111 iii mmm mm bc u abc u ab −−−    % % 1 2 33 ii R R uR −− = ## Consider the linearized advection-diffusion equation. 2 2 uu Cv tx ∂∂∂ += u x discretized with the backward Euler method, and second order spatial differences. 1 2 2 2 nn C uuu v 1 n i x ++ + +− =− ∆∆ −+ + + 1 + There are three unknowns. ,, nnn Write system in the form [] [] [] i Au Bu Cu R R +++ ++= = 1 22 Rearranging gives 2 1 n i Ct vt vt u xx 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 21 1 0, or 0 uu u x and with the definitions that 1 2 and gives for 0 Ct vt u xx µβ ∆∆ = . 1 1 1 2 1 1 1 10 12 22 1 2 0 1 n n n m n m u u u u µµ ββ β + + + +   −− + +− +  % # % # 2 1 1 0 0 What would first line look like for 0? n n m u u u x = = # # The cases in HW #2 have much more complicated R.H.S. vectors.
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CFD lecture 4 - Computational Fluid Dynamics Lecture 4...

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