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Empirical versus Direct sensitivity computation.pdf - SimHydro 2010

2D **sensitivity** equations5∂s∂t + ∂G x∂x+ ∂G y∂y= Q⎛s ≡ ∂U ∂h/∂ψ∂ψ = ⎝ ∂q/∂ψ∂r/∂ψ⎞⎠ =⎛⎝ η θρ⎞⎠G x = ∂F x∂U s, G y = ∂F y∂U sQ = ∂S∂U s + ∂S∂ψ ε − ∂∂x( ∂Fx∂ψ ε )− ∂ ∂y( ∂Fy∂ψ ε )Guinot et al. Adv. in Water Resources (31), 2009Delenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

Agenda**Empirical** vs **Direct** approaches6✦3 examples of **sensitivity** analysis (SA):✦✦✦1D steady flow on a sloping bed (backwater curve)1D transient flow with discontinuous solution (dambreak)2D flow in a roundaboutDelenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

1D Steady flow on a sloping bed 7SA with respect to the friction coefficient n✦ Reference solution:The backwater curve and its derivative with respect to n(discretization using Explicit Euler method)dηdhdx = S 0 − S f1 − Fr 2dx = η ∂∂h( )S0 − S f1 − Fr 2( )∂ S0 − S f+ ε nM∂n M 1 − Fr 2!'()*!"#&!!"+, -./ 0 1# 2!$"!"#$%+,-,.,/0,!#"%&'&(&)*&!"#$1!()*!"3!+,2✦ Steady state => q=constant and θ=0Delenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

1D Steady flow on a sloping bedSA with respect to the friction coefficient n8✦Numerical flow solution✦✦Resolution of the 1d SWEs with HLLC Riemann solverTransient simulation until the steady state is reachedDelenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

1D Steady flow on a sloping bedSA with respect to the friction coefficient n8✦Numerical flow solution✦✦Resolution of the 1d SWEs with HLLC Riemann solverTransient simulation until the steady state is reached!'()*!"#&!"#$%+,-,.,/0,12),.3045!2!3. 4 567!%#""%!%!"#$$$&'(')'*+',-.')/+01!"#$6!()*8!3.7Incorrect estimation of q due to the balancing process in the HLLC(flux at interfaces and values at cells)Delenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

1D Steady flow on a sloping bed 9✦ Sensitivity solution:SA with respect to the friction coefficient n!!"%& '() * +# ,!$"!#"./0/1/23/4&5616378!"!$"-!%&,!#"%&'&(&)*&+,(&*-!".!/01Equivalent results for ηDelenne et al. Num. Meth. in Fluids, in revisionDelenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

1D Discontinuous flow: Dambreak10SA with respect to the initial water depth in the dam h up✦ Initial condition problemq(x, 0) = 0h(x, 0) ={h up x

1D Discontinuous flow 12SA with respect to the initial water depth in the dam h up✦ **Empirical** method: compute the hydraulic solution with 2slightly different values of h uphh uph up +dh up∆hdh up≤ 1Delenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

1D Discontinuous flow 12SA with respect to the initial water depth in the dam h up✦ **Empirical** method: compute the hydraulic solution with 2slightly different values of h uphh uph up +dh upProblem across the shock:Δh remains high whendh up tends to 0∆hdh up≤ 1Delenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

1D Discontinuous flow 13SA with respect to the initial water depth in the dam h up✦**Direct** approach:✦Sensitivity equations defined for continuous solutionsModification: extra source term R applied only in case of shock∂s∂t + ∂G∂x = Q + Rδ sIn case of a Riemann problem, R can be simplified intoR = − ∂c s∂ψ (U L − U R )‣ U L , U R : left and rightstates of the Riemannproblem‣ c s : shock speedDelenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

1D Discontinuous flow 14SA with respect to the initial water depth in the dam h up✦**Direct** approach:✦Solve the **sensitivity** equations(including R) with thefollowing initial conditions:η(x, 0) =θ(x, 0) = 0{1 x

2D flow in a roundaboutSA with respect to the boundary condition in street C: h C15q upq upABDh DCh C0m10mDelenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

2D flow in a roundabout 16SA with respect to the boundary condition in street C: h C✦Flow solution:✦Solve the 2D SWEs with the HLLC Riemann solverwater depth hdischarge (q,r)Delenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

2D flow in a roundabout 16SA with respect to the boundary condition in street C: h C✦Flow solution:✦Solve the 2D SWEs with the HLLC Riemann solverwater depth hdischarge (q,r)Delenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

2D flow in a roundaboutSA with respect to the boundary condition in street C: h C17✦Sensitivity η of h with respect to hC**Empirical****Direct**Guinot et al. Adv. in Water Resources (32), 2009Delenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

2D flow in a roundabout18✦ Sensitivity θ of q with respect to hCSA with respect to the boundary condition in street C: h C**Empirical****Direct**Guinot et al. Adv. in Water Resources (31), 2009Delenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

Conclusion 19**Empirical**Applicable even if the modelequations are not knownvs**Direct**Requires the model equationsand their derivative / ψTwo simulationsOne simulationProblem with discontinuousflows (shock, jump...)Possible discontinuous flowsPropagates the artifacts of flowsolution to the **sensitivity**Eliminates those artifactsDelenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

Conclusion 19**Empirical**Applicable even if the modelequations are not knownvs**Direct**Requires the model equationsand their derivative / ψTwo simulationsOne simulationProblem with discontinuousflows (shock, jump...)Possible discontinuous flowsPropagates the artifacts of flowsolution to the **sensitivity**Eliminates those artifactsDelenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

Conclusion 19**Empirical**Applicable even if the modelequations are not knownvs**Direct**Requires the model equationsand their derivative / ψTwo simulationsOne simulationProblem with discontinuousflows (shock, jump...)Possible discontinuous flowsPropagates the artifacts of flowsolution to the **sensitivity**Eliminates those artifactsDelenne, Finaud‐Guyot, Guinot, Cappelaere ‐ **Empirical** **versus** **Direct** **sensitivity** **computation**s ‐ **SimHydro** **2010**.

**Empirical** **versus** **Direct****sensitivity** **computation**Application to the shallow water equationsC. Delenne ‐ P. Finaud‐Guyot ‐ V. Guinot ‐ B. CappelaereHydroSciences Montpellier ‐ UMR 5569 ‐ CNRS IRD UM1 UM2Simhydro **2010**, 2‐4 June **2010**, Sophia Antipolis