Finite Difference Solutions to the ADE Simplest form

Finite Difference Solutions to the ADE

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

Effect of Numerical Errors (overshoot) (MT 3 DMS manual)

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

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

Example from Zheng &Bennett v = 100 cm/h l = 100 cm C 1= 100 mg/l C 2= 10 mg/l With no dispersion, breakthrough occurs at t = v/ l = 1 hour

Explicit approximation with upstream weighting v = 100 cm/hr l = 100 cm C 1= 100 mg/l C 2= 10 mg/l t = 0. 1 hr

Implicit Approximations Implicit; upstream weighting Implicit; central differences

= Finite Element Method

Governing Equation for Ogata and Banks solution

x x j-1/2 j j+1/2

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

Stability Constraints for the 1 D 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

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

Spreadsheet solution (on course homepage)

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

x x j-1/2 j j+1/2

In general: Upstream weighting: See equations 7. 11 and 7. 17 in Zheng & Bennett