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Finite Difference Solutions to the ADE

Finite Difference Solutions to the ADE. Even Simpler form. Simplest form of the ADE. Plug Flow Plug Source. Flow Equation. Effect of Numerical Errors. (overshoot). (MT3DMS manual).  x. v. j. j+1. j-1. x. Explicit approximation with upstream weighting.

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Finite Difference Solutions to the ADE

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  1. Finite Difference Solutions to the ADE

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

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

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

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

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

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

  9. = Finite Element Method

  10. Governing Equation for Ogata and Banks solution

  11. j-1/2 j+1/2 x x j j+1 j-1 Central difference approximation

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

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

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

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

  16. Co j+1 j+1 j-1 j j-1 j Boundary Conditions Specified concentration boundary a “free mass outflow” boundary (Z&B, p. 285) Cb= Co Cb= Cj

  17. Spreadsheet solution (on course homepage) Co Specified concentration boundary a “free mass outflow” boundary Cb= Cj Cb= Co

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

  19. j-1/2 j+1/2 x x j j+1 j-1

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

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