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PETE 603

PETE 603. Lecture Session #28 Tuesday, 7/27/10. Direct/Iterative Methods. Iterative methods (systems of linear equations) Computer time increases more linearly with the number of unknowns Convergence criteria must be considered. CPU Time. Iterative. Direct. Number of Unknowns.

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PETE 603

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  1. PETE 603 Lecture Session #28 Tuesday, 7/27/10

  2. Direct/Iterative Methods • Iterative methods (systems of linear equations) • Computer time increases more linearly with the number of unknowns • Convergence criteria must be considered CPU Time Iterative Direct Numberof Unknowns

  3. Gauss-Seidel Iteration

  4. Gauss-Seidel Iteration

  5. Gauss-Seidel Iteration

  6. Gauss-Seidel Iteration

  7. Gauss-Seidel Iteration

  8. Gauss-Seidel Iteration

  9. Gauss-Seidel Iteration

  10. Gauss-Seidel Iteration

  11. 3 6 9 12 2 5 8 11 1 4 7 10 Gauss-Seidel Iteration Application to Reservoir Simulation: Consider this system and the 2-D finite difference equation Ap = r We can write the equation for Block 5 and for a general block

  12. Gauss-Seidel Iteration Rearrangement gives: Point Successive Overrelaxation (PSOR)

  13. LSOR Iteration Again, consider this system and the 2-D finite difference equation Ap = r Now write all of the equations for the 2nd column (first column unknowns already updated):

  14. Gauss-Seidel Iteration These can be written Use the latest iteration for the East and West pressures, and solve using the Thomas Algorithm This is Line Successive Overrelaxation (LSOR)

  15. 6 12 18 24 4 10 16 22 2 8 14 20 5 11 17 23 3 9 15 21 1 7 13 19 Gauss-Seidel Iteration Now, consider this system and the 3-D finite difference equation Ap = r Now, the equations for each layer or slice of rows or columns can be written, and at each iteration, a 2-D matrix equation must be solved. This is Block Successive Overrelaxation (BSOR). Top Bottom

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