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Numerical Hydraulics
Shallow water equations in 1D:Method of characteristics
Wolfgang Kinzelbach withMarc Wolf andCornel Beffa
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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
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• 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
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• 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
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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
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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
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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.
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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
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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
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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
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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