1. Finite Difference Solutions to the ADE

2. Simplest form of the ADE Even Simpler form Plug Flow Plug Source Flow Equation

3. Effect of Numerical Errors (overshoot) (MT3DMS manual)

4. (See Zheng & Bennett, p. 174-181) v j-1 jj+1 xx x Explicit approximation with upstream weighting

5. Explicit; Upstream weighting (See Zheng & Bennett, p. 174-181) v j-1 jj+1 xx x

6. Example from Zheng &Bennett v = 100 cm/h  l = 100 cm C1= 100 mg/l C2= 10 mg/l With no dispersion, breakthrough occurs at t = v/  l = 1 hour

7. v = 100 cm/hr  l = 100 cm C1= 100 mg/l C2= 10 mg/l  t = 0.1 hr Explicit approximation with upstream weighting

8. Implicit; central differences Implicit; upstream weighting Implicit Approximations

10. = Finite Element Method

11. Governing Equation for Ogata and Banks solution

12. j-1 jj+1 xx x j-1/2j+1/2

13. Governing Equation for Ogata and Banks solution Finite difference formula: explicit with upstream weighting, assuming v >0 Solve for c j n+1

14. Stability Constraints for the 1D Explicit Solution (Z&B, equations 7.15, 7.16, 7.36, 7.40) Courant Number Cr < 1 Stability Criterion Peclet Number Controls numerical dispersion & oscillation, see Fig.7.5

15. CoCo Boundary Conditions a “free mass outflow” boundary (Z&B, p. 285) Specified concentration boundary C b = C o C b = C j j j+1 j-1j j+1 j-1

16. Spreadsheet solution (on course homepage)

17. We want to write a general form of the finite difference equation allowing for either upstream weighting (v either + or –) or central differences.

18. j-1 jj+1 xx x j-1/2j+1/2

19. Upstream weighting: In general: See equations 7.11 and 7.17 in Zheng & Bennett