Finite Difference Solutions to the ADE

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

Effect of Numerical Errors (overshoot) (MT3DMS manual)

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

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

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

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

Implicit; central differences Implicit; upstream weighting Implicit Approximations

= Finite Element Method

Governing Equation for Ogata and Banks solution

j-1 jj+1 xx x j-1/2j+1/2 Central difference approximation

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

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

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.

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

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

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

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

j-1 jj+1 xx x j-1/2j+1/2

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