Finite Difference Solutions

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Finite Difference Solutions to the ADE
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Simplest form of the ADE
Plug Flow Plug Source
Flow Equation
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Effect of Numerical Errors
(overshoot)
(MT3DMS manual)
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(See Zheng Bennett, p. 174-181)
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v
Explicit Upstream weighting
(See Zheng Bennett, p. 174-181)
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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
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v 100 cm/hr ?l 100 cm C1 100 mg/l C2 10
mg/l ?t 0.1 hr
Explicit approximation with upstream weighting
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Implicit Approximations
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(No Transcript)
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Finite Element Method
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Governing Equation for Ogata and Banks solution
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x
?x
j
j1
j-1
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Governing Equation for Ogata and Banks solution
Finite difference formula explicit with upstream
weighting, assuming v gt0
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Stability Constraints for the 1D Explicit
Solution (ZB, equations 7.15, 7.16, 7.36, 7.40)
Courant Number
Stability Criterion
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Boundary Conditions
Specified concentration boundary
a free mass outflow boundary (ZB, p. 285)
Cb Co
Cb Cj
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Spreadsheet solution (on course homepage)
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We want to write a general form of the finite
difference equation allowing for either upstream
weighting (v either or ) or central
differences.
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x
?x
j
j1
j-1
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In general
Upstream weighting
See equations 7.11 and 7.17 in Zheng Bennett