CFD lecture 14 - Computational Fluid Dynamics Lecture 14...

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Computational Fluid Dynamics Lecture 14 Iterative Methods: Consider the 1-D analog: () 2 2 11 2 2 2 22 ii i i i p Rx x pp p R x i p px p +− = −+ = +∆ =− R Jacobi Method: 12 1 2 nn n p pp x R + =+ Here n means the iteration level, not the time step. 1 + Gauss-Siedel Method: 1 2 n i 2 i p x R ++ Uses updated ( values as soon as they become available. ) 2 1 n + Gauss-Siedel is typically twice as fast as Jacobi iteration, but can’t be accelerated with over- relaxation. S.O.R. – Successive over relaxation Relaxation parameter 0 λ << necessary for convergence. 2 1 1 2 n i i n i p x R p λλ  +   Finding the right value of is important to the rate of convergence. There is an optimal value, but finding it is as expensive as solving the original problem. Typical values near 0.5 or 1.5 == 1
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If 01 λ << then the current value becomes a weighted average of the Gauss-Siedel value and the value from the previous iteration. This is termed under-relaxation .
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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 14 - Computational Fluid Dynamics Lecture 14...

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