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Efficient Techniques for Implicit 1D & 2D Approximation Solutions

Learn how to solve implicit approximation problems using point iteration, matrix solutions, and a combination of both techniques. This approach is utilized in MODFLOW, offering efficient solutions for complex equations. Discover effective methods like ADI and tridiagonal matrix solutions. Examples include IADI, SSOR, and SIP techniques.

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Efficient Techniques for Implicit 1D & 2D Approximation Solutions

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  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. Solve by iteration or 1D Implicit Approximation

  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. Tridiagonal solution oriented along columns 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

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

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