This is a draft presentation with the first application of the BEq model ... now it is published on http://onlinelibrary.wiley.com/doi/10.1002/wrcr.20072/abstract

1. Emanuele Cordano
Double-grid 2D solver for
Bossinesq Equation
Friday, June 7, 2013

2. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
2
Boussinesq Equation for subsurface flow (theory)
s
∂η
∂t
= ∇ · [KS H(η, x, y) ∇η] + Q
2D Boussinesq Equation: water mass balance in a point + Darcy’s law
Symbol Dimension Definition
s, s(x,y) porosity
t [T] time
x,y [L] space (planimetric) coordinate
H,H(x, y, η) [L] water-table thickness
KS [L T -1
] saturated hydraulic conductivity
Q [L T -1
] source term (water-table discharge)
η, η(x, y, t) [L] water surface elevation in a generic point (x, y, t)
Friday, June 7, 2013

3. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
3
Boussinesq Equation for subsurface flow (theory)
s
∂η
∂t
= ∇ · [KS H(η, x, y) ∇η] + Q
accumulation term: time derivative of water content
Symbol Dimension Definition
s, s(x,y) porosity
t [T] time
x,y [L] space (planimetric) coordinate
H,H(x, y, η) [L] water-table thickness
KS [L T -1
] saturated hydraulic conductivity
Q [L T -1
] source term (water-table discharge)
η, η(x, y, t) [L] water surface elevation in a generic point (x, y, t)
Friday, June 7, 2013

4. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
4
Boussinesq Equation for subsurface flow (theory)
sfrac {partial eta} {partial t} = nabla cdot left[ K_S , H (eta,x,y) , nabla eta right]+Q
s
∂η
∂t
= ∇ · [KS H(η, x, y) ∇η] + Q
(-) divergence of water flux (vector) described by Darcy’law
Symbol Dimension Definition
s, s(x,y) porosity
t [T] time
x,y [L] space (planimetric) coordinate
H,H(x, y, η) [L] water-table thickness
KS [L T -1
] saturated hydraulic conductivity
Q [L T -1
] source term (water-table discharge)
η, η(x, y, t) [L] water surface elevation in a generic point (x, y, t)
Friday, June 7, 2013

5. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
5
Boussinesq Equation for subsurface flow (theory)
sfrac {partial eta} {partial t} = nabla cdot left[ K_S , H (eta,x,y) , nabla eta right]+Q
s
∂η
∂t
= ∇ · [KS H(η, x, y) ∇η] + Q
source term (water-table dicharge due to rainfall or sinks, etc.
Symbol Dimension Definition
s, s(x,y) porosity
t [T] time
x,y [L] space (planimetric) coordinate
H,H(x, y, η) [L] water-table thickness
KS [L T -1
] saturated hydraulic conductivity
Q [L T -1
] source term (water-table discharge)
η, η(x, y, t) [L] water surface elevation in a generic point (x, y, t)
Friday, June 7, 2013

6. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
6
Boussinesq Equation for subsurface flow (theory)
sfrac {partial eta} {partial t} = nabla cdot left[ K_S , H (eta,x,y) , nabla eta right]+Q
s
∂η
∂t
= ∇ · [KS H(η, x, y) ∇η] + Q
2D Boussinesq Equation: water mass balance in a point + Darcy’s law
Symbol Dimension Definition
s, s(x,y) porosity
t [T] time
x,y [L] space (planimetric) coordinate
H,H(x, y, η) [L] water-table thickness
KS [L T -1
] saturated hydraulic conductivity
Q [L T -1
] source term (water-table discharge)
η, η(x, y, t) [L] water surface elevation in a generic point (x, y, t)
Friday, June 7, 2013

7. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
7
Vi(ηn+1
i ) − ∆tn
j∈Si
KS Aj(ηn
i,m(i,j))
ηn+1
m(i,j) − ηn+1
i
δi,m(i,j)
= Vi(ηn
i ) + ∆ tn
Qn+1
i pi
Discretized Boussinesq Equation (implicit)
water volume at the next time step (unknown)
Symbol Dimension Definition
l(j) index of the cell on the left-hand side of the j-th line segment
m(i,j) index of the cell that shares the j-th line segment with the i-th cell
pi [L2
] topographic area of the i-th cell
r(j) index of the cell on the right-hand side of the j-th line segment
Ai(ηr(j),l(j)) [L2
] Vertical area over the j-th line
Qn
i [L T -1
] source term (water-table discharge) at the i-th cell in the n-th time instant
Si * set of the edges of the i-th cell
Vi(ηi) [L3
] water volume stored in the i-th cell defined as a function of water surface elevation ηi
V(η) [L3
] vectors of water volume stored each cell defined as a function of vector of water surface elevation η
δiq [L] euclidean distance between the centroids of the i-th and q-th cells
ηi [L] water surface elevation at the i-th cell
ηi,d [L] averaged water surface elevation between the i-th and d-th cells
ηn
i [L] water surface elevation at the i-th cell in the n-th time instant
∆tn
[T] time step between the n-th and the (n+1)-th time instants
Friday, June 7, 2013

8. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
8
Vi(ηn+1
i ) − ∆tn
j∈Si
KS Aj(ηn
i,m(i,j))
ηn+1
m(i,j) − ηn+1
i
δi,m(i,j)
= Vi(ηn
i ) + ∆ tn
Qn+1
i pi
Discretized Boussinesq Equation (implicit)
net out-going fluxes (unknown) derived by discretized Darcy’s Law
Symbol Dimension Definition
l(j) index of the cell on the left-hand side of the j-th line segment
m(i,j) index of the cell that shares the j-th line segment with the i-th cell
pi [L2
] topographic area of the i-th cell
r(j) index of the cell on the right-hand side of the j-th line segment
Ai(ηr(j),l(j)) [L2
] Vertical area over the j-th line
Qn
i [L T -1
] source term (water-table discharge) at the i-th cell in the n-th time instant
Si * set of the edges of the i-th cell
Vi(ηi) [L3
] water volume stored in the i-th cell defined as a function of water surface elevation ηi
V(η) [L3
] vectors of water volume stored each cell defined as a function of vector of water surface elevation η
δiq [L] euclidean distance between the centroids of the i-th and q-th cells
ηi [L] water surface elevation at the i-th cell
ηi,d [L] averaged water surface elevation between the i-th and d-th cells
ηn
i [L] water surface elevation at the i-th cell in the n-th time instant
∆tn
[T] time step between the n-th and the (n+1)-th time instants
Friday, June 7, 2013

9. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
9
Vi(ηn+1
i ) − ∆tn
j∈Si
KS Aj(ηn
i,m(i,j))
ηn+1
m(i,j) − ηn+1
i
δi,m(i,j)
= Vi(ηn
i ) + ∆ tn
Qn+1
i pi
Discretized Boussinesq Equation (implicit)
water volume at the previous time step (known)
Symbol Dimension Definition
l(j) index of the cell on the left-hand side of the j-th line segment
m(i,j) index of the cell that shares the j-th line segment with the i-th cell
pi [L2
] topographic area of the i-th cell
r(j) index of the cell on the right-hand side of the j-th line segment
Ai(ηr(j),l(j)) [L2
] Vertical area over the j-th line
Qn
i [L T -1
] source term (water-table discharge) at the i-th cell in the n-th time instant
Si * set of the edges of the i-th cell
Vi(ηi) [L3
] water volume stored in the i-th cell defined as a function of water surface elevation ηi
V(η) [L3
] vectors of water volume stored each cell defined as a function of vector of water surface elevation η
δiq [L] euclidean distance between the centroids of the i-th and q-th cells
ηi [L] water surface elevation at the i-th cell
ηi,d [L] averaged water surface elevation between the i-th and d-th cells
ηn
i [L] water surface elevation at the i-th cell in the n-th time instant
∆tn
[T] time step between the n-th and the (n+1)-th time instants
Friday, June 7, 2013

10. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
10
Vi(ηn+1
i ) − ∆tn
j∈Si
KS Aj(ηn
i,m(i,j))
ηn+1
m(i,j) − ηn+1
i
δi,m(i,j)
= Vi(ηn
i ) + ∆ tn
Qn+1
i pi
Discretized Boussinesq Equation (implicit)
source term (in-coming water volume during the time-step) (known)
Symbol Dimension Definition
l(j) index of the cell on the left-hand side of the j-th line segment
m(i,j) index of the cell that shares the j-th line segment with the i-th cell
pi [L2
] topographic area of the i-th cell
r(j) index of the cell on the right-hand side of the j-th line segment
Ai(ηr(j),l(j)) [L2
] Vertical area over the j-th line
Qn
i [L T -1
] source term (water-table discharge) at the i-th cell in the n-th time instant
Si * set of the edges of the i-th cell
Vi(ηi) [L3
] water volume stored in the i-th cell defined as a function of water surface elevation ηi
V(η) [L3
] vectors of water volume stored each cell defined as a function of vector of water surface elevation η
δiq [L] euclidean distance between the centroids of the i-th and q-th cells
ηi [L] water surface elevation at the i-th cell
ηi,d [L] averaged water surface elevation between the i-th and d-th cells
ηn
i [L] water surface elevation at the i-th cell in the n-th time instant
∆tn
[T] time step between the n-th and the (n+1)-th time instants
Friday, June 7, 2013

11. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
11
Vi(ηn+1
i ) − ∆tn
j∈Si
KS Aj(ηn
i,m(i,j))
ηn+1
m(i,j) − ηn+1
i
δi,m(i,j)
= Vi(ηn
i ) + ∆ tn
Qn+1
i pi
Discretized Boussinesq Equation (implicit)
Water volume budget for each cell of the domain (algebaraic equation system)
Symbol Dimension Definition
l(j) index of the cell on the left-hand side of the j-th line segment
m(i,j) index of the cell that shares the j-th line segment with the i-th cell
pi [L2
] topographic area of the i-th cell
r(j) index of the cell on the right-hand side of the j-th line segment
Ai(ηr(j),l(j)) [L2
] Vertical area over the j-th line
Qn
i [L T -1
] source term (water-table discharge) at the i-th cell in the n-th time instant
Si * set of the edges of the i-th cell
Vi(ηi) [L3
] water volume stored in the i-th cell defined as a function of water surface elevation ηi
V(η) [L3
] vectors of water volume stored each cell defined as a function of vector of water surface elevation η
δiq [L] euclidean distance between the centroids of the i-th and q-th cells
ηi [L] water surface elevation at the i-th cell
ηi,d [L] averaged water surface elevation between the i-th and d-th cells
ηn
i [L] water surface elevation at the i-th cell in the n-th time instant
∆tn
[T] time step between the n-th and the (n+1)-th time instants
Friday, June 7, 2013

12. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
12
V(ηn+1
) + T · ηn+1
= b3
b3i = Vi(ηn
i ) + ∆ tn
Qn+1
i pi
T · ηn+1
i
= −∆tn
j∈Si
KS Aj(ηn
i,m(i,j))
ηn+1
m(i,j) − ηn+1
i
δi,m(i,j)
Symbol Dimension Definition
b3 [L3
] vector of known terms
T [L2
] symmetric matrix introduced in the equation systems derived by Darcy’s law
V(η) [L3
] vectors of water volume stored each cell defined as a function of vector of water surface e
Nonlinear Algebraic Equation System
Friday, June 7, 2013

13. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
13
m+1
ηn+1
=m
ηn+1
−
Ws(m
ηn+1
) + T
−1
V(m
ηn+1
) + T ·m
ηn+1
− b3
Wsi(ψi) =
dVi(ηi)
dηi
Nonlinear Algebraic Equation System (2)
•Resolution with a Newton-like iterative scheme (Casulli,2008):
Symmetric and Positive-Defined Matrix which is solved with
the Conjugate Gradient Method
where
Friday, June 7, 2013

14. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
14
Algorithm of the Conjugate Gradient Method from “ A
Introduction to the Conjugate Gradient Method Without the
Agonizing Pain” by Jonathan Richard Shewchuk, 1994
(http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)
Friday, June 7, 2013

15. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
15
Boussinesq Equation for subsurface flow:
Results of two simulations of water drainege in a
mountain catchment
Friday, June 7, 2013

16. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
16
A “drying” study case: Matsch Valley,
(bottom elevation [m] at 20 m resolution
Friday, June 7, 2013

17. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
17
Initial Conditions and Parameters
(*) water table 1 m over the averaged elevation of the cell of the coarse
grid (in this case averaged on the 60 m resolution)
No rainfall, impermeable boundary condition
Index kS [m/s] Porosity Basin Code Cell Res. [m] ∆t [s] Initial Conditions
1 10−1
0.4 Matsch v2.1.4 60 104
1 m of w.th.(*)
2 10−1
0.4 Matsch v3.1 60 104
1 m of w.th.(*)
Code according to Brugnano Casulli, 2008 (v2.1.4) , general case (v3.1)
Friday, June 7, 2013

18. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
18
Water Table Thickness (case 1)
[cm]
time 10^4 s time 10^5 s time 10^6 s time 5 10^6 s
Friday, June 7, 2013

19. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
19
Water Table Thickness (case 2)
[cm]
time 10^4 s time 10^5 s time 10^6 s time 5 10^6 s
Friday, June 7, 2013

20. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
20
Resume
Index Stored Water Volume [m3
] Global Relative Error
1 3.94 · 107
10−7
2 3.94 · 107
10−7
Friday, June 7, 2013

21. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
21
Analysis of the number of iterations: CG and NP vs time
0e+00 1e+06 2e+06 3e+06 4e+06
050010001500200025003000
Max CG Iterations in a time step
time
0e+00 1e+06 2e+06 3e+06 4e+06
0246810
Newton/Picard Iteration in a time step
time
Newton-PicardIterationinatimestep
v2.1.4 (blue) , v3.1 (red)
Friday, June 7, 2013

22. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
22
0 500 1000 1500 2000 2500 3000
05001000150020002500
Max Iteration of Conjugate Gradient Method (v3.1 vs v2.1.4)
Max CG Iterations in v2.4.1
MaxCGIterationsinv3.1
Analysis of the number of CG iterations: v2.1.4 vs v3.1
Friday, June 7, 2013

23. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
23
Work in Progress: simulation with v3.1
Index kS [m/s] Porosity Basin FG Res.[m] CG Res. [m] Initial Conditions
3 10−1
0.4 Matsch 20 60 1 m of w.th. over CG (averaged)
Water thickness [cm] at 10^6 s
Too many CG iterations: more
than 2000 per each time step
Friday, June 7, 2013

24. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
24
Preconditioner from “ An Introduction to the Conjugate Gradient
Method Without the Agonizing Pain” by Jonathan Richard
Shewchuk, 1994
(http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)
conditioning number
number of CG iteration
tollerance
Friday, June 7, 2013

25. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
25
Preconditioner (algorithm): http://www.cs.cmu.edu/~quake-
papers/painless-conjugate-gradient.pdf
Friday, June 7, 2013

26. Double-grid 2D solver for Boussinesq Equation
Emanuele Cordano
26
Major details about the numerical theory with newer applications are now
discussed by Cordano and Rigon, 2013
Other Numerical references:
Casulli, V. (2009), A high-resolution wetting and drying algorithm for freesurface
hydrodynamics, Int. J. Numer. Methods Fluids, 60(4), 391–408,
doi:10.1002/fld.1896.
Brugnano, L., and V. Casulli (2008), Iterative solution of piecewise linear
systems, J. Sci. Comput., 30(1), 463–472.
Friday, June 7, 2013