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Numerical Hydraulics

Shallow water equations in 1D:Method of characteristics

Wolfgang Kinzelbach withMarc Wolf andCornel Beffa

Characteristic equations

• As shown before, the St. Venant equations can be put into the form:

• To obtain total differentials in the brackets we have to choose

S E

v v h g hv h v g I I

t x t x

1,2

g gB

h A

• Thus we obtain the characteristic equations:

along

along

S E

Dv g Dhg I I

Dt c Dt

S E

Dv g Dhg I I

Dt c Dt

dxv c

dt

dxv c

dt

• Positive and negative characteristics for sub-critical, critical and super-critical flow:

Types of characteristics

t t tPP P

C+ C+C+

C-

C-C-

EE EW W Wx x x

Terminology: P, W (West) und E (East) instead of i, i-1, i+1

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Integration of the characteristic equations

• Multiplication with dt and integration– along characteristic line

– and along characteristic line

– yields:

P

W

ES

P

W

P

W

dtIIgdhc

gdv

P

E

ES

P

E

P

E

dtIIgdhc

gdv

dxv c

dt

dxv c

dt

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Integration of the characteristic equations

WPWESWPW

WP ttIIghhc

gvv

EPEESEPE

EP ttIIghhc

gvv

PWpP hCCv

P n E Pv C C h

WPWESWW

Wp ttIIghc

gvC

EPEESEE

En ttIIghc

gvC

( / ) ( / )E E W WC g c C g c

or

This implies a linearisation. The wavevelocity becomes constant in the element.

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Grid for subcritical flow (1)

Zeit

x

j

j+1P

W E

Characteristics start on grid points

Terminology C (Center), W (West) und E (East) instead of i, i-1, i+1

C

Problem: Characteristicsintersect between grid pointsin points P at time levels which do not coincide withthe time levels of the grid. Results have to be interpolated.

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Grid for subcritical flow (2)

Zeit

x

j

j+1

P

W E

Characteristic lines end at point P, starting points do not coincide withgrid points. Values at starting points are obtained by interpolation from grid point values

Terminology C (Center), W (West) und E (East) instead of i, i-1, i+1

C

We choose thisvariant!

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Interpolation

x

tcv

xx

xx

xx

xx

vv

vv LL

WC

LP

WC

LC

WC

LC

L LC L

C W

v c tc c

c c x

Solve for two unknowns vL and cL

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Starting point L

• Solution for vL and cL yields:

WCWC

WCCWC

L

ccvvx

t

vcvcx

tv

v

1

WC

WCLC

L

ccx

t

ccx

tvc

c

1

WCLLCL hhcvx

thh

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Starting point R for subcritical flow

• In analogy to point L, variables for point R vR and cR

ECEC

ECCEC

R

ccvvxt

vcvcxt

vv

1

EC

ECRC

R

ccxt

ccxt

vcc

1

ECRRCR hhcvx

thh

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Starting point R for supercritical flow

• Starting point of characteristic between W and C

• Using velocity v-c we obtain

WCWC

WCCWC

R

ccvvx

t

vcvcx

tv

v

1

CW

WCRC

R

ccx

t

ccx

tvc

c

1

WCRRCR hhcvx

thh

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Final explicit working equations

tIIghc

gvC LESL

LLp

tIIghc

gvC RESR

RRn

P p L Pv C C h P n R Pv C C h with

and

( / ) ( / )L L R RC g c C g c

Integration from L to P and from R to P

2 equations with2 unknowns

Boundary conditions are required as discussedin FD method.As method is explicitCLF-criterium applies.

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Classical dam break problem:Solution with method of

characteristics (Test problem)

Propagation velocity of fronts slightly too high

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Matrix form of the St. Venant equations (1D)

0u f

bt x

2 20

( )2

s E

qh

u f bq gh gh I Iqh

Finite volume method

• FV formulation for this vector equation:

• e and w designate the east and west boundary of the FV cell respectively

0u f

bt x

1t t e w

tu u f f b t

x

ii-1 i+1ew

x

Finite volume method

• The computation of the term

can be done in different ways. E.g. with an upwind scheme (e becomes i and w becomes i-1 if the wave propagates in positive x-direction.)

• For the time discretisation we choose an explicit method

e wf f

Upwind formulation

1

2 2 2 21 1

1

,0 ,0

,0 ,02 2

t ti i

t tt t t te i ii i i i

t t t ti i i i

q q

f q qq gh q gh

q h q h

1

2 2 2 21 1

1

,0 ,0

,0 ,02 2

t ti i

t tt t t tw i ii i i i

t t t ti i i i

q q

f q qq gh q gh

q h q h

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Improvement of method

• A further improvement can be reached by flux-limiting

• The Roe-method is such an improvement. It can take into account discontinuities across the cell boundaries.

Flux difference splitting (Roe)

• Idea: At the cell boundary the flux is computed according to the characteristics by a positive/negative linear wave (splitting).

• The flux at the east side of a cell is:

( )e l r lf f A u u

( )e r l rf f A u u ii-1 i+1

l r

orew

x

Flux difference splitting (Roe)

• The Roe matrix A is the Jacobian matrix of the flux vector

• The division into left and right part allows to account for discontinuities.

f

fA

u

The Roe matrix

• The Roe matrix can be computed as:

11 12

21 22

1

2

a aA

a ac

11 1 2 1 2 12 1 2

21 1 2 2 1 22 1 1 2 2

a a

a a

1

qc

h 2

qc

h

Fluxes according to Roe scheme

• The fluxes at a cell side are computed from the left/right-side fluxes, e.g.:

• and the variables on the new time level are:

1 1

2 2e e e l rf f f A u u

( ) ( )i i e w

tu t t u t f f b

x

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Advection: Shallow water equations:

Fluxes:

0u bt x

1

( ) 0t ti i

e wu u bt

0u f

bt x

1

( ) 0t ti i

e w

u uf f b

t

1

1( )

2e i iu u u central

( )e i

u u upwind

( )e i

f f upwind

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Shallow water equations with first order upwind (Flux from left/west):

11

t t t t ti i i i i

th h q q b

x

2 2 2 21

12 2

t t

t t ti i i

i i

t q gh q ghq q b

x h h