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VORtechComputing Delft University of Technology
Improving convergence ofQuickFlowA Steady State Solver for the ShallowWater Equations
Femke van Wageningen-Kessels
January 22, 2007
January 22, 2007 2
VORtechComputing Delft University of Technology
Background
January 22, 2007 3
VORtechComputing Delft University of Technology
Background: software
WAQUA QuickFlow- since 1970’s - - since 2006 -
• slow convergence• time-dependent
solution• not well suited for
stationary solution
• fast convergence• only stationary
solution• uses WAQUA output
as initial solution
January 22, 2007 4
VORtechComputing Delft University of Technology
Goal & content
• How does QuickFlow work?• Shallow water equations• Spatial discretization• Time iteration methods• Test problems
• Improvements to QuickFlow• Planning
January 22, 2007 5
VORtechComputing Delft University of Technology
2D Shallow water equations
Navier Stokes equations ⇒ Reynolds equations⇒ 2D Shallow water equations:
∂U∂t
+ U ∂U∂x
+ V ∂U∂y
− fV + g∂ζ∂x
−τbottom,x
ρ0H− νH
(
∂2U∂x2
+∂2U∂y2
)
= 0
∂V∂t
+ U ∂V∂x
+ V ∂V∂y
+ fU + g∂ζ∂y
−τbottom,y
ρ0H− νH
(
∂2V∂x2
+∂2V∂y2
)
= 0
∂ζ∂t
+∂HU∂x
+∂HV∂y
= 0
⇒ Shallow water equations on curvilinear grid
January 22, 2007 6
VORtechComputing Delft University of Technology
Spatial discretization: grid
PSfrag replacements
11
2
2
3
3
4
5
m
n depthwater level U -velocityV -velocity
identical (m, n) index
one computational cell
January 22, 2007 7
VORtechComputing Delft University of Technology
Spatial discretization: methods
Momentum equations:
• Third order upwind: U ∂V∂x
and V ∂U∂y
• Central difference
Continuity equations:• Finite volume
January 22, 2007 8
VORtechComputing Delft University of Technology
Spatial discretization: boundaries
PSfrag replacements
water levelboundary
discharge orvelocity
boundary
closedfree slip
January 22, 2007 9
VORtechComputing Delft University of Technology
Spatial discretization: moving boundary
PSfrag replacementsboundary
Hsurface
bottom
H ≤ δ ⇒ U = 0
January 22, 2007 10
VORtechComputing Delft University of Technology
Time iteration: WAQUA - ADI
stage 1: V -momentum explicitdudt
= A1(ul,ul+1/2)
stage 2: U -momentum & continuity explicitdudt
= A2(ul+1/2,ul+1)
Stationary:dudt
= 1
2[A1(u,u) + A2(u,u)] ≡ A(u)
January 22, 2007 11
VORtechComputing Delft University of Technology
Time iteration: Euler Backward
du
dt= A(u) ⇒
un+1 − u
n
∆t= A(un+1) ⇒
∆t M (un+1− u
n) + A(un+1) − b = 0
Advantages:• easy implementation• unconditionally stable
January 22, 2007 12
VORtechComputing Delft University of Technology
Time iteration: Newtons method
F(x) ≡ ∆tM(un+1− u
n) + A(un+1) − b = 0
xk+1 = x
k− α
(
F′(xk)
)−1F(xk)
Advantages:• easy implementation• global convergence (line search)
January 22, 2007 13
VORtechComputing Delft University of Technology
Test problem: Chézy
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.0
0.1
0.2
0.3
0.4
0.5
X−coordinate [km]
Y−
coor
dina
te [k
m]
Solution by WAQUA
[m]
0.2
0.4
0.6
0.8
1
1.2
• Rectangular gutter
• 25 × 11 grid cells
• inflow:velocity boundary
• outflow:water level boundary
January 22, 2007 14
VORtechComputing Delft University of Technology
Test problems: real
Lek Randwijk
January 22, 2007 15
VORtechComputing Delft University of Technology
Test problem: Lek
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
−0.5
0.0
0.5
1.0
1.5
2.0
X−coordinate [km]
Y−
coor
dina
te [k
m]
MAPS.sep at T=Inf, stationary
[m]
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
−0.5
0.0
0.5
1.0
1.5
2.0
X−coordinate [km]
Y−
coor
dina
te [k
m]
Difference WAQUA − QuickFlow
[m]
−7
−6
−5
−4
−3
−2
−1
0
1
2
x 10−3
• Relatively easy
• 42 × 34 grid cells
• inflow:discharge boundary
• outflow:water level boundary
January 22, 2007 16
VORtechComputing Delft University of Technology
Test problem: Randwijk
170.0 175.0 180.0 185.0 190.0432.0
434.0
436.0
438.0
440.0
442.0
444.0
446.0
448.0
450.0
X−coordinate [km]
Y−
coor
dina
te [k
m]
Difference WAQUA − QuickFlow
[m]
−0.025
−0.02
−0.015
−0.01
−0.005
170.0 175.0 180.0 185.0 190.0432.0
434.0
436.0
438.0
440.0
442.0
444.0
446.0
448.0
450.0
X−coordinate [km]
Y−
coor
dina
te [k
m]
Difference WAQUA − QuickFlow
[m]
−0.025
−0.02
−0.015
−0.01
−0.005
• Relatively complex
• 287 × 26 grid cells
• inflow:discharge boundary
• outflow:water level boundary
January 22, 2007 17
VORtechComputing Delft University of Technology
Improvements: damping BC’s
’hard’ open boundaries ⇒
nonphysical reflection of waves⇒ long ’run-up’ time
Remedie:damping boundary conditions
January 22, 2007 18
VORtechComputing Delft University of Technology
Improvements: semi-explicit time
Example: advection equation∂u∂t
= u∂u∂x
Euler Backward & upwindun+1
m − unm
∆t= un+1
mun+1
m − unm−1
∆x
Alternativeun+1
m − unm
∆t= un
mun+1
m − unm−1
∆x
Implementation in Euler Backward or in Newton
January 22, 2007 19
VORtechComputing Delft University of Technology
Improvements: quasi-Newton
Approximate Jacobian F′(x0) ≈ Bk
xk+1 = x
k−(
Bk)−1
F(
xk)
Advantages:• reduce costs for computing F′
• reduce costs for LU-decomposition of F′
• possibly faster convergence (depends onmethod)
January 22, 2007 20
VORtechComputing Delft University of Technology
Improvements: adaptive time stepping
complicated area→ small time step
or large time step for continuity equation:
ζ l+1 − ζ l
∆t+
∂
∂x(Hu) = 0
∆t → ∞ ⇒∂
∂x(Hu) = 0
January 22, 2007 21
VORtechComputing Delft University of Technology
Plan
• Resolve large errors near open boundaries• Inventorise & solve problems
• Chézy• Lek• Randwijk
January 22, 2007 22
VORtechComputing Delft University of Technology
Conclusion
• Software: WAQUA & QuickFlow• Modelling river flow: Shallow water equations• Spatial discretization• Time iteration: ADI, Euler Backward, Newton• Improvements:
• Damping boundary conditions• Semi-explicit time iteration• Quasi-Newton• Adaptive time stepping
January 22, 2007 23
VORtechComputing Delft University of Technology
Questions?