1. Implicit approximation can be solved using: Point iteration (G/S, SOR) Direct (matrix) solution Combination of matrix soln and iteration (used in MODFLOW)

2. Implicit Approximation or 1D Solve by iteration

3. In this form, the equation can be solved directly using matrix methods. See W&A, p. 95. All known terms are on the RHS; all unknown terms are on the LHS.

4. Let  x =  y = a 2D

5. The motivation behind the Alternating Direction Implicit Procedure is to keep the coefficient matrix tridiagonal so that we can use the Thomas algorithm to solve the matrix equation. Not tridiagonal Tridiagonal solution oriented along columns

6. Tridiagonal solution oriented along rows In the next time step, the solution is oriented along rows.

7. In point iteration, the 5-point operator moves over each node in the grid….

8. In the ADI matrix solution, the 5-point equations are assembled into one matrix equation for each column (or row).

9. Examples of solution techniques that combine matrix solution with iteration: IADI (see chapter 5 of W&A) SSOR* SIP* PCG2* *Used in MODFLOW