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

2. 28.1 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 Number of Unknowns Direct Iterative

3. 28.2 Gauss-Seidel Iteration

4. 28.3 Gauss-Seidel Iteration

5. 28.4 Gauss-Seidel Iteration

6. 28.5 Gauss-Seidel Iteration

7. 28.6 Gauss-Seidel Iteration

8. 28.7 Gauss-Seidel Iteration

9. 28.8 Gauss-Seidel Iteration

10. 28.9 Gauss-Seidel Iteration

11. 28.10 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 3 6 9 12 2 5 8 11 1 4 7 10

12. 28.11 Gauss-Seidel Iteration Rearrangement gives: Point Successive Overrelaxation (PSOR)

13. 28.12 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. 28.13 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. 28.14 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). 6 12 18 24 4 10 16 22 2 8 14 20 Top 5 11 17 23 3 9 15 21 1 7 13 19 Bottom