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 =  l/v = 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 Central difference approximation

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 Criterion for Explicit Approximation For dispersion alone For advection alone (Courant number) For both

15. Stability Constraints for the 1D Explicit Solution (Z&B, equations 7.15, 7.16, 7.36, 7.40) Courant Number Cr < 1 Stability Criterion Also need to minimize numerical dispersion.

16. Numerical Dispersion controlled by the Courant Number and the Peclet Number for all numerical solutions (both explicit and implicit approximations) Courant NumberCr < 1 Peclet Number Controls numerical dispersion & oscillation, see Fig.7.5

17. 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

18. Spreadsheet solution (on course homepage) CoCo a “free mass outflow” boundary Specified concentration boundary C b = C o C b = C j

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

20. j-1 jj+1 xx x j-1/2j+1/2

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