CE 498/698 and ERS 685 (Spring 2004)

Lecture 13 1

Lecture 13: Advanced Time-Variable Solutions

CE 498/698 and ERS 685

Principles of Water Quality Modeling

CE 498/698 and ERS 685 (Spring 2004)

Lecture 13 2

Lecture 13

• Advanced time-variable solutions– Implicit approaches >1 unknown in ea. eqn.

only 1 unknown in ea. eqn.– Explicit approaches

CE 498/698 and ERS 685 (Spring 2004)

Lecture 13 3

Lecture 13

• Advanced time-variable solutions– Implicit approaches– Explicit approaches

derivation of methodsnot covered inthis class

know for this class• Table 13.1

CE 498/698 and ERS 685 (Spring 2004)

Lecture 13 4

Recall from Lecture 12Constraints:

Numerical dispersion: 2

5.02 tU

xUEn

Stability:

2

2

2 xkExU

xt

Finite difference approaches:

FTCS 5.0

FTBS 1explicit methods

CE 498/698 and ERS 685 (Spring 2004)

Lecture 13 5

Lecture 13

• Explicit method– MacCormack – CTCS

• Implicit methods– BTCS– Crank-Nicholson

CE 498/698 and ERS 685 (Spring 2004)

Lecture 13 6

DifficultNoneCrank-

Nicholson

DifficultBTCS

ModerateConditional (more liberal)NoneMacCormack

CTCS

EasyFTCS

EasyFTBS

Implemen-tation Ease

StabilityNumerical dispersion

Method

tU

xUEn

25.0

5.0 2

2

2 xkExU

xt

tUEn 25.0

22

2 xkE

xt

tUEn 25.0 t

t

Table 13.1

error inerror inbook!book!