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# Double-grid 2D solver for Boussinesq Equation (BEq) ... Draft

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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

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### Double-grid 2D solver for Boussinesq Equation (BEq) ... Draft

1. 1. Emanuele CordanoDouble-grid 2D solver forBossinesq EquationFriday, June 7, 2013
2. 2. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano2Boussinesq Equation for subsurface flow (theory)s∂η∂t= ∇ · [KS H(η, x, y) ∇η] + Q2D Boussinesq Equation: water mass balance in a point + Darcy’s lawSymbol Dimension Deﬁnitions, s(x,y) porosityt [T] timex,y [L] space (planimetric) coordinateH,H(x, y, η) [L] water-table thicknessKS [L T -1] saturated hydraulic conductivityQ [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. 3. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano3Boussinesq Equation for subsurface flow (theory)s∂η∂t= ∇ · [KS H(η, x, y) ∇η] + Qaccumulation term: time derivative of water contentSymbol Dimension Deﬁnitions, s(x,y) porosityt [T] timex,y [L] space (planimetric) coordinateH,H(x, y, η) [L] water-table thicknessKS [L T -1] saturated hydraulic conductivityQ [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. 4. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano4Boussinesq Equation for subsurface flow (theory)sfrac {partial eta} {partial t} = nabla cdot left[ K_S , H (eta,x,y) , nabla eta right]+Qs∂η∂t= ∇ · [KS H(η, x, y) ∇η] + Q(-) divergence of water flux (vector) described by Darcy’lawSymbol Dimension Deﬁnitions, s(x,y) porosityt [T] timex,y [L] space (planimetric) coordinateH,H(x, y, η) [L] water-table thicknessKS [L T -1] saturated hydraulic conductivityQ [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. 5. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano5Boussinesq Equation for subsurface flow (theory)sfrac {partial eta} {partial t} = nabla cdot left[ K_S , H (eta,x,y) , nabla eta right]+Qs∂η∂t= ∇ · [KS H(η, x, y) ∇η] + Qsource term (water-table dicharge due to rainfall or sinks, etc.Symbol Dimension Deﬁnitions, s(x,y) porosityt [T] timex,y [L] space (planimetric) coordinateH,H(x, y, η) [L] water-table thicknessKS [L T -1] saturated hydraulic conductivityQ [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. 6. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano6Boussinesq Equation for subsurface flow (theory)sfrac {partial eta} {partial t} = nabla cdot left[ K_S , H (eta,x,y) , nabla eta right]+Qs∂η∂t= ∇ · [KS H(η, x, y) ∇η] + Q2D Boussinesq Equation: water mass balance in a point + Darcy’s lawSymbol Dimension Deﬁnitions, s(x,y) porosityt [T] timex,y [L] space (planimetric) coordinateH,H(x, y, η) [L] water-table thicknessKS [L T -1] saturated hydraulic conductivityQ [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. 7. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano7Vi(ηn+1i ) − ∆tnj∈SiKS Aj(ηni,m(i,j))ηn+1m(i,j) − ηn+1iδi,m(i,j)= Vi(ηni ) + ∆ tnQn+1i piDiscretized Boussinesq Equation (implicit)water volume at the next time step (unknown)Symbol Dimension Deﬁnitionl(j) index of the cell on the left-hand side of the j-th line segmentm(i,j) index of the cell that shares the j-th line segment with the i-th cellpi [L2] topographic area of the i-th cellr(j) index of the cell on the right-hand side of the j-th line segmentAi(ηr(j),l(j)) [L2] Vertical area over the j-th lineQni [L T -1] source term (water-table discharge) at the i-th cell in the n-th time instantSi * set of the edges of the i-th cellVi(ηi) [L3] water volume stored in the i-th cell deﬁned as a function of water surface elevation ηiV(η) [L3] vectors of water volume stored each cell deﬁned as a function of vector of water surface elevation ηδiq [L] euclidean distance between the centroids of the i-th and q-th cellsηi [L] water surface elevation at the i-th cellηi,d [L] averaged water surface elevation between the i-th and d-th cellsηni [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 instantsFriday, June 7, 2013
8. 8. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano8Vi(ηn+1i ) − ∆tnj∈SiKS Aj(ηni,m(i,j))ηn+1m(i,j) − ηn+1iδi,m(i,j)= Vi(ηni ) + ∆ tnQn+1i piDiscretized Boussinesq Equation (implicit)net out-going fluxes (unknown) derived by discretized Darcy’s LawSymbol Dimension Deﬁnitionl(j) index of the cell on the left-hand side of the j-th line segmentm(i,j) index of the cell that shares the j-th line segment with the i-th cellpi [L2] topographic area of the i-th cellr(j) index of the cell on the right-hand side of the j-th line segmentAi(ηr(j),l(j)) [L2] Vertical area over the j-th lineQni [L T -1] source term (water-table discharge) at the i-th cell in the n-th time instantSi * set of the edges of the i-th cellVi(ηi) [L3] water volume stored in the i-th cell deﬁned as a function of water surface elevation ηiV(η) [L3] vectors of water volume stored each cell deﬁned as a function of vector of water surface elevation ηδiq [L] euclidean distance between the centroids of the i-th and q-th cellsηi [L] water surface elevation at the i-th cellηi,d [L] averaged water surface elevation between the i-th and d-th cellsηni [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 instantsFriday, June 7, 2013
9. 9. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano9Vi(ηn+1i ) − ∆tnj∈SiKS Aj(ηni,m(i,j))ηn+1m(i,j) − ηn+1iδi,m(i,j)= Vi(ηni ) + ∆ tnQn+1i piDiscretized Boussinesq Equation (implicit)water volume at the previous time step (known)Symbol Dimension Deﬁnitionl(j) index of the cell on the left-hand side of the j-th line segmentm(i,j) index of the cell that shares the j-th line segment with the i-th cellpi [L2] topographic area of the i-th cellr(j) index of the cell on the right-hand side of the j-th line segmentAi(ηr(j),l(j)) [L2] Vertical area over the j-th lineQni [L T -1] source term (water-table discharge) at the i-th cell in the n-th time instantSi * set of the edges of the i-th cellVi(ηi) [L3] water volume stored in the i-th cell deﬁned as a function of water surface elevation ηiV(η) [L3] vectors of water volume stored each cell deﬁned as a function of vector of water surface elevation ηδiq [L] euclidean distance between the centroids of the i-th and q-th cellsηi [L] water surface elevation at the i-th cellηi,d [L] averaged water surface elevation between the i-th and d-th cellsηni [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 instantsFriday, June 7, 2013
10. 10. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano10Vi(ηn+1i ) − ∆tnj∈SiKS Aj(ηni,m(i,j))ηn+1m(i,j) − ηn+1iδi,m(i,j)= Vi(ηni ) + ∆ tnQn+1i piDiscretized Boussinesq Equation (implicit)source term (in-coming water volume during the time-step) (known)Symbol Dimension Deﬁnitionl(j) index of the cell on the left-hand side of the j-th line segmentm(i,j) index of the cell that shares the j-th line segment with the i-th cellpi [L2] topographic area of the i-th cellr(j) index of the cell on the right-hand side of the j-th line segmentAi(ηr(j),l(j)) [L2] Vertical area over the j-th lineQni [L T -1] source term (water-table discharge) at the i-th cell in the n-th time instantSi * set of the edges of the i-th cellVi(ηi) [L3] water volume stored in the i-th cell deﬁned as a function of water surface elevation ηiV(η) [L3] vectors of water volume stored each cell deﬁned as a function of vector of water surface elevation ηδiq [L] euclidean distance between the centroids of the i-th and q-th cellsηi [L] water surface elevation at the i-th cellηi,d [L] averaged water surface elevation between the i-th and d-th cellsηni [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 instantsFriday, June 7, 2013
11. 11. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano11Vi(ηn+1i ) − ∆tnj∈SiKS Aj(ηni,m(i,j))ηn+1m(i,j) − ηn+1iδi,m(i,j)= Vi(ηni ) + ∆ tnQn+1i piDiscretized Boussinesq Equation (implicit)Water volume budget for each cell of the domain (algebaraic equation system)Symbol Dimension Deﬁnitionl(j) index of the cell on the left-hand side of the j-th line segmentm(i,j) index of the cell that shares the j-th line segment with the i-th cellpi [L2] topographic area of the i-th cellr(j) index of the cell on the right-hand side of the j-th line segmentAi(ηr(j),l(j)) [L2] Vertical area over the j-th lineQni [L T -1] source term (water-table discharge) at the i-th cell in the n-th time instantSi * set of the edges of the i-th cellVi(ηi) [L3] water volume stored in the i-th cell deﬁned as a function of water surface elevation ηiV(η) [L3] vectors of water volume stored each cell deﬁned as a function of vector of water surface elevation ηδiq [L] euclidean distance between the centroids of the i-th and q-th cellsηi [L] water surface elevation at the i-th cellηi,d [L] averaged water surface elevation between the i-th and d-th cellsηni [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 instantsFriday, June 7, 2013
12. 12. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano12V(ηn+1) + T · ηn+1= b3b3i = Vi(ηni ) + ∆ tnQn+1i piT · ηn+1i= −∆tnj∈SiKS Aj(ηni,m(i,j))ηn+1m(i,j) − ηn+1iδi,m(i,j)Symbol Dimension Deﬁnitionb3 [L3] vector of known termsT [L2] symmetric matrix introduced in the equation systems derived by Darcy’s lawV(η) [L3] vectors of water volume stored each cell deﬁned as a function of vector of water surface eNonlinear Algebraic Equation SystemFriday, June 7, 2013
13. 13. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano13m+1ηn+1=mηn+1−Ws(mηn+1) + T−1 V(mηn+1) + T ·mηn+1− b3Wsi(ψi) =dVi(ηi)dηiNonlinear Algebraic Equation System (2)•Resolution with a Newton-like iterative scheme (Casulli,2008):Symmetric and Positive-Defined Matrix which is solved withthe Conjugate Gradient MethodwhereFriday, June 7, 2013
14. 14. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano14Algorithm of the Conjugate Gradient Method from “ AIntroduction to the Conjugate Gradient Method Without theAgonizing Pain” by Jonathan Richard Shewchuk, 1994(http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)Friday, June 7, 2013
15. 15. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano15Boussinesq Equation for subsurface flow:Results of two simulations of water drainege in amountain catchmentFriday, June 7, 2013
16. 16. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano16A “drying” study case: Matsch Valley,(bottom elevation [m] at 20 m resolutionFriday, June 7, 2013
17. 17. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano17Initial Conditions and Parameters(*) water table 1 m over the averaged elevation of the cell of the coarsegrid (in this case averaged on the 60 m resolution)No rainfall, impermeable boundary conditionIndex kS [m/s] Porosity Basin Code Cell Res. [m] ∆t [s] Initial Conditions1 10−10.4 Matsch v2.1.4 60 1041 m of w.th.(*)2 10−10.4 Matsch v3.1 60 1041 m of w.th.(*)Code according to Brugnano Casulli, 2008 (v2.1.4) , general case (v3.1)Friday, June 7, 2013
18. 18. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano18Water Table Thickness (case 1)[cm]time 10^4 s time 10^5 s time 10^6 s time 5 10^6 sFriday, June 7, 2013
19. 19. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano19Water Table Thickness (case 2)[cm]time 10^4 s time 10^5 s time 10^6 s time 5 10^6 sFriday, June 7, 2013
20. 20. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano20ResumeIndex Stored Water Volume [m3] Global Relative Error1 3.94 · 10710−72 3.94 · 10710−7Friday, June 7, 2013
21. 21. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano21Analysis of the number of iterations: CG and NP vs time0e+00 1e+06 2e+06 3e+06 4e+06050010001500200025003000Max CG Iterations in a time steptime0e+00 1e+06 2e+06 3e+06 4e+060246810Newton/Picard Iteration in a time steptimeNewton-PicardIterationinatimestepv2.1.4 (blue) , v3.1 (red)Friday, June 7, 2013
22. 22. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano220 500 1000 1500 2000 2500 300005001000150020002500Max Iteration of Conjugate Gradient Method (v3.1 vs v2.1.4)Max CG Iterations in v2.4.1MaxCGIterationsinv3.1Analysis of the number of CG iterations: v2.1.4 vs v3.1Friday, June 7, 2013
23. 23. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano23Work in Progress: simulation with v3.1Index kS [m/s] Porosity Basin FG Res.[m] CG Res. [m] Initial Conditions3 10−10.4 Matsch 20 60 1 m of w.th. over CG (averaged)Water thickness [cm] at 10^6 sToo many CG iterations: morethan 2000 per each time stepFriday, June 7, 2013
24. 24. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano24Preconditioner from “ An Introduction to the Conjugate GradientMethod Without the Agonizing Pain” by Jonathan RichardShewchuk, 1994(http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)conditioning numbernumber of CG iterationtolleranceFriday, June 7, 2013
25. 25. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano25Preconditioner (algorithm): http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdfFriday, June 7, 2013
26. 26. Double-grid 2D solver for Boussinesq EquationEmanuele Cordano26Major details about the numerical theory with newer applications are nowdiscussed by Cordano and Rigon, 2013Other Numerical references:Casulli, V. (2009), A high-resolution wetting and drying algorithm for freesurfacehydrodynamics, Int. J. Numer. Methods Fluids, 60(4), 391–408,doi:10.1002/fld.1896.Brugnano, L., and V. Casulli (2008), Iterative solution of piecewise linearsystems, J. Sci. Comput., 30(1), 463–472.Friday, June 7, 2013