Morphodynamics Relaxation solver Stochastic aspects Numerics for Hydromorphodynamics Processes Relaxation Solvers and Stochastic Aspects E. Audusse . LAGA, UMR 7569, Univ. Paris 13 ANGE group (CETMEF – INRIA – UPMC - CNRS) August 7, 2015 E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Saint-Venant – Exner Model : A CEMRACS Story ◮ CEMRACS 2011 : Numerical Simulation Sediment transport modelling : Relaxation schemes for Saint-Venant Exner and three layer models. Emmanuel Audusse, Christophe Berthon, Christophe Chalons, Olivier Delestre, Nicole Goutal, Magali Jodeau, Jacques Sainte-Marie, Jan Giesselmann and Georges Sadaka. ESAIM Proc., Vol. 38, pp 78-98, 2012. ◮ CEMRACS 2013 : Stochastic Aspects Numerical simulation of the dynamics of sedimentary river beds with a stochastic Exner equation. Emmanuel Audusse, S´ ebastien Boyaval, Nicole Goutal, Magali Jodeau and Philippe Ung. ESAIM Proc., Vol. 48, pp 312-340, 2015. E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Morphodynamic processes Coastal erosion River Morphodynamics Dunes formation Soil erosion Dam drain Industrial sites E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Morphodynamics processes . Morphodynamics process Suspended- and bedload E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Modelling for bedload transport ◮ Saint-Venant – Exner model ◮ Implemented at the industrial level ◮ Empirical derivation, no energy ◮ Reasonable results for a large class of experiences ◮ A lot of parameters to tune... ◮ Two-layer (Saint-Venant ?) models ◮ Well-known in the hyperbolic community for bi-fluid modelling ◮ Validity of the extension to bedload transport ? ◮ Definition of the layers ? ◮ Interfaces conditions ? ◮ Rheology law in the solid part ? E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Saint-Venant – Exner model ◮ Equations ∂ t h + ∂ x q w = 0 , � q 2 � − gh ∂ x b − τ b h + g w 2 h 2 ∂ t q w + ∂ x = , ρ w ρ s (1 − p ) ∂ t b + ∂ x q s = 0 , (Exner[25], Paola-Voller[05], VanRijn[93,06], Parker[06], Cordier[11], Garegnani[13]...) ◮ Comments ◮ Fluid quantities : Water depth ( h ) and discharge ( q w ) ◮ Sediment quantity : Position of the interface ( b ) ◮ 2 conservation equations + 1 dynamic equation E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Saint-Venant – Exner model ◮ Equations ∂ t h + ∂ x q w = 0 , � q 2 � − gh ∂ x b − τ b h + g w 2 h 2 ∂ t q w + ∂ x = , ρ w ρ s (1 − p ) ∂ t b + ∂ x q s = 0 , (Exner[25], Paola-Voller[05], VanRijn[93,06], Parker[06], Cordier[11], Garegnani[13]...) ◮ Comments ◮ No dynamics in the solid phase ◮ No sediment transport in the fluid phase ◮ Closure relations for friction term τ b and sediment flux q s E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Friction coefficient formulae ◮ Laminar flow τ b = κ ( h ) u Gerbeau[01] ◮ Engineers formulae τ b = κ ( h ) u 2 ◮ Chezy formula κ ( h ) = κ Formule pour trouver la vitesse uniforme que l’eau aura dans un foss´ e ou dans un canal dont la pente est connue. Applications pour la Seine et l’Yvette [Chezy, 1776] ◮ Manning formula κ κ ( h ) = h 1 / 3 Manning[1891] E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Sediment flux formulae ◮ Power laws (Exner[25], Grass[81]) u | m ¯ q s = A g | ¯ u ◮ Shields parameter τ b = u 2 | τ b | /ρ w θ = , s = ρ s /ρ w , Kh α g ( s − 1) d m ◮ Threshold sediment flux formulae � g ( s − 1) d 3 q s = Φ m uller [48] : Φ = 8 ( θ − θ c ) 3 / 2 ◮ Meyer-Peter & M¨ + ◮ Engelund & Fredsoe [76]: Φ = 18 . 74( θ − θ c ) + ( θ 1 / 2 − 0 . 7 θ 1 / 2 ) c ◮ Nielsen [92] : Φ = 12 θ 1 / 2 ( θ − θ c ) + ... E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects ”Steady” flow over a movable bump Dunes Antidunes Fluvial flow Torrential flow E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects ”Steady” flow over a movable bump in 2d (O. Delestre, CEMRACS 2012) E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Numerical strategies for bedload transport ◮ Steady state strategy ◮ Hydrodynamic computation on fixed topography � Steady state ◮ Evolution of topography forced by hydrodynamic steady state ◮ Efficient for phenomena involving very different time scales ◮ External coupling (time splitting) ◮ Use of two different softwares for hydro- and morphodynamics ◮ Allow to use existing solvers and different numerical strategies ◮ Actual strategy in industrial softwares ◮ Efficient for low coupling (?) ◮ Internal coupling ◮ Solution of the whole system at once ◮ Need for a new solver ◮ Efficient for general coupling (Hudson[05], Castro[08], Delis[08], Benkhaldoun[09], Murillo[10]...) E. Audusse Numerics for hydromorphodynamics
Morphodynamics Relaxation solver Stochastic aspects Dam break : Failure for time splitting External coupling (M. Jodeau, EDF) E. Audusse Numerics for hydromorphodynamics
Morphodynamics Scalar conservation law Relaxation solver SW system and Suliciu approaches Stochastic aspects SW-Exner system and Suliciu extensions Finite Volume Method f : R p → R 2 × p u ∈ R p , ∂ t u + ∇ · f ( u ) = 0 , ◮ Integration on the prism C i × [ t n , t n +1 ] � t n +1 � � � u ( t n +1 , x ) dx = u ( t n , x ) dx − � f ( u ( t , s )) . n ij dtds t n C i C i Γ ij j ∈ V ( i ) with � t n +1 1 � F ( U n i , U n j , n ij ) ≈ f ( u ( t , s )) . n ij dtds ∆ t n | Γ ij | t n Γ ij E. Audusse Numerics for hydromorphodynamics
Morphodynamics Scalar conservation law Relaxation solver SW system and Suliciu approaches Stochastic aspects SW-Exner system and Suliciu extensions Finite Volume Method f : R p → R 2 × p u ∈ R p , ∂ t u + ∇ · f ( u ) = 0 , ◮ Integration on the prism C i × [ t n , t n +1 ] ij = ∆ t n | Γ ij | U n +1 � = U n σ n ij F ( U n i , U n σ n i − j , n ij ) , i | C i | j ∈ V ( i ) with � t n +1 1 � F ( U n i , U n j , n ij ) ≈ f ( u ( t , s )) . n ij dtds ∆ t n | Γ ij | Γ ij t n E. Audusse Numerics for hydromorphodynamics
Morphodynamics Scalar conservation law Relaxation solver SW system and Suliciu approaches Stochastic aspects SW-Exner system and Suliciu extensions Godunov scheme ◮ Riemann problem � u l x ≤ 0 if ∂ t u + ∂ x f ( u ) = 0 , u (0 , x ) = u r if x > 0 � Much more easy to solve that the general IBVP ◮ Godunov scheme ◮ Start from piecewise constant initial data ◮ Solve the Riemann pb at each interface x i +1 / 2 : u r i +1 / 2 ( t , x ) ◮ Fix the time step so that the Riemann problems do not interact ◮ Construct a global solution by merging the solutions of all the local Riemann problems u g ( t , x ) = u r i +1 / 2 ( t , x ) if x ∈ [ x i , x i +1 ] ◮ Take the meanvalue of this solution at time ∆ t n � x i +1 / 2 1 U n +1 u g (∆ t n , x ) dx = i ∆ x i x i − 1 / 2 E. Audusse Numerics for hydromorphodynamics
Morphodynamics Scalar conservation law Relaxation solver SW system and Suliciu approaches Stochastic aspects SW-Exner system and Suliciu extensions Godunov scheme ◮ Riemann problem � u l x ≤ 0 if ∂ t u + ∂ x f ( u ) = 0 , u (0 , x ) = u r if x > 0 � Much more easy to solve that the general IBVP ◮ Godunov scheme ◮ Start from piecewise constant initial data ◮ Solve the Riemann pb at each interface x i +1 / 2 : u r i +1 / 2 ( t , x ) ◮ Fix the time step so that the Riemann problems do not interact ◮ Construct a global solution by merging the solutions of all the local Riemann problems u g ( t , x ) = u r i +1 / 2 ( t , x ) if x ∈ [ x i , x i +1 ] ◮ For conservative equations, equivalent with FV approach � t n 1 u r F i +1 / 2 = i +1 / 2 ( t , x i +1 / 2 ) dt ∆ t n 0 E. Audusse Numerics for hydromorphodynamics
Morphodynamics Scalar conservation law Relaxation solver SW system and Suliciu approaches Stochastic aspects SW-Exner system and Suliciu extensions Relaxation solver: General idea ◮ Godunov scheme ◮ Consistency, Stability ◮ Complexity : Solution of the Riemann problem (rarefaction waves, shocks, contact discontinuities...) ◮ Relaxation models : Introduction of a larger system ◮ that is hyperbolic (with LD fields) ◮ that formally converges to the physical one ◮ that ensures some stability properties ◮ for which the (homogeneous) Riemann problem is easy to solve ◮ Numerical algorithm ◮ Definition of auxiliary variables from physical ones ◮ Solution of the (homogeneous) Riemann problem ◮ Computation of the physical variables at the next time step (Suliciu [92], Chen et al. [94], Jin-Xin [95], Nonlinear Stability of FVM for Hyperbolic Conservation Laws [Bouchut, 04]...) E. Audusse Numerics for hydromorphodynamics
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