From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) A relaxation framework for morphodynamics modelling E. Audusse . LAGA, UMR 7569, Univ. Paris 13 BANG project-team, INRIA Paris-Rocquencourt . HYP 2012 - Padova June 26, 2012 E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Joint work with ◮ BANG project - ANGE group M.O. Bristeau, J. Sainte-Marie ◮ EDF LNHE – Saint-Venant Lab. N. Goutal, M. Jodeau ◮ C. Berthon, C. Chalons, O. Delestre, S. Cordier E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) From SW flows to Morphodynamics Shallow Water Flows Morphodynamic processes Relaxation framework for SW-Exner model Relaxation model Relaxation scheme Numerical results Towards new models (and other perspectives) E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Shallow Water Flows Relaxation framework for SW-Exner model Morphodynamic processes Towards new models (and other perspectives) Shallow water flows ◮ Shallow water equations ∂ t h + ∇ · ( h u ) = 0 , h u ⊗ u + gh 2 � � ∂ t ( h u ) + ∇ · 2 I = − gh ∇ z − 2Ω × h u − κ ( h , u ) u ◮ Applications E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Shallow Water Flows Relaxation framework for SW-Exner model Morphodynamic processes Towards new models (and other perspectives) Shallow water flows ◮ Shallow water equations ∂ t h + ∇ · ( h u ) = 0 , h u ⊗ u + gh 2 � � ∂ t ( h u ) + ∇ · 2 I = − gh ∇ z − 2Ω × h u − κ ( h , u ) u ◮ Positive and well-balanced numerical schemes ◮ Extended Godunov schemes (Greenberg-Leroux) ◮ Kinetic interpretation of source terms (Pertame-Simeoni) ◮ Extended Suliciu relaxation schemes (Bouchut) ◮ Hydrostatic reconstruction (ABBKP) ◮ Hydrostatic upwind (Berthon-Foucher) ◮ ... E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Shallow Water Flows Relaxation framework for SW-Exner model Morphodynamic processes Towards new models (and other perspectives) From SW flows to coupled problems ◮ Pollutant processes INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF 6.5 6.5 11.5 11.5 .0325 .0325 .0975 .0975 4 4 9 9 INRIA-MODULEF INRIA-MODULEF 14 14 0 0 .065 .065 INRIA-MODULEF INRIA-MODULEF .13 .13 ◮ Hydrobiological processes (A.C. Boulanger) Depth of particles through time. Omega = 1.33. 0.05 Water height 0 Particle 1 Particle 2 Particle 3 −0.05 −0.1 −0.15 Depth (m) −0.2 −0.25 −0.3 −0.35 −0.4 −0.45 0 500 1000 1500 2000 2500 3000 3500 Time(s) E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Shallow Water Flows Relaxation framework for SW-Exner model Morphodynamic processes Towards new models (and other perspectives) Morphodynamic processes ◮ Dune formation ◮ Coastal erosion ◮ Impact on industrial building (harbour, dam, nuclear plant...) ◮ River morphodynamic ◮ Strong events (tsunami, dam drain or break...) E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Shallow Water Flows Relaxation framework for SW-Exner model Morphodynamic processes Towards new models (and other perspectives) Hyperbolic models in Morphodynamics ◮ Suspended sediment model ◮ Applications : High coupling and light sediments ◮ SW equations + Transport + Bottom evolution (ODE) ◮ Closure : Erosion and deposition source terms ◮ Bedload transport model ◮ Applications : Low coupling or heavy sediments ◮ SW equations + Bottom evolution ◮ Closure : Sediment flux ◮ Empirical formula : Grass, Meyer-Peter-M¨ uller, Einstein... E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Shallow Water Flows Relaxation framework for SW-Exner model Morphodynamic processes Towards new models (and other perspectives) 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 low coupling and different time scales (dune formation) ◮ External coupling ◮ Use of two different softwares for hydro- and morphodynamics ◮ Allow to use existing solvers and different numerical strategies ◮ Actual strategy at EDF (MASCARET-COURLIS) ◮ Efficient for low coupling (slow river morphodynamics) ◮ Internal coupling ◮ Solution of the whole system at once ◮ Need for a new solver ◮ Efficient for high coupling (dam drain, tsunami) E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Shallow Water Flows Relaxation framework for SW-Exner model Morphodynamic processes Towards new models (and other perspectives) Saint-Venant – Exner model ∂ H ∂ t + ∂ Q = 0 , ∂ x � Q 2 � ∂ Q ∂ t + ∂ H + g − gH ∂ Z 2 H 2 = ∂ x , ∂ x ρ s (1 − p ) ∂ Z ∂ t + ∂ Q s = 0 , ∂ x ◮ Hyperbolic for classical choices of Q s ( h , u ) ◮ Eigenvalues hard to compute except for special choices of Q s ◮ No dynamic effects in the solid phase ◮ No transport in the fluid phase ◮ Numerical strategies : Hudson, Nieto, Morales, Benkhaldoun... E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Relaxation model Relaxation framework for SW-Exner model Relaxation scheme Towards new models (and other perspectives) Numerical results Relaxation model ∂ t H + ∂ x Hu = 0 Hu 2 + Π � � ∂ t Hu + ∂ x = − gH ∂ x Z ∂ t Π + u ∂ x Π + a 2 � gH 2 1 � = − Π H ∂ x u λ 2 ∂ t Z + ∂ x Ω = 0 � b 2 � 1 H 2 − u 2 ∂ t Ω + ∂ x Z + 2 u ∂ x Ω = λ ( Q s − Ω) ◮ (Π , Ω) : Auxiliary variables (fluid pressure, sediment flux) ◮ λ > 0 : (Small) relaxation parameter ◮ ( a , b ) > 0 : Have to be fixed to ensure stability E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Relaxation model Relaxation framework for SW-Exner model Relaxation scheme Towards new models (and other perspectives) Numerical results Relaxation model ∂ t H + ∂ x Hu = 0 Hu 2 + Π � � ∂ t Hu + ∂ x = − gH ∂ x Z ∂ t Π + u ∂ x Π + a 2 � gH 2 1 � = − Π H ∂ x u λ 2 ∂ t Z + ∂ x Ω = 0 � b 2 � 1 H 2 − u 2 ∂ t Ω + ∂ x Z + 2 u ∂ x Ω = λ ( Q s − Ω) ◮ (Π , Ω) : Auxiliary variables (fluid pressure, sediment flux) ◮ λ > 0 : (Small) relaxation parameter ◮ ( a , b ) > 0 : Have to be fixed to ensure stability E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Relaxation model Relaxation framework for SW-Exner model Relaxation scheme Towards new models (and other perspectives) Numerical results Main properties ◮ Formally tends to SW-Exner model when λ tends to 0 ◮ No explicit dependency on sediment flux Q S ◮ Always hyperbolic ( H � = 0) ◮ Eigenvalues easy to compute (case a < b ) u − b H < u − a H < u < u + a H < u + b H ◮ Linearly degenerate system � Exact (homogeneous) Riemann problem ”easy” to solve E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Relaxation model Relaxation framework for SW-Exner model Relaxation scheme Towards new models (and other perspectives) Numerical results Stability of the relaxation model ◮ Chapman-Enskog expansion Π = p ( h ) + λ Π 1 + ... Ω = Q s ( h , u ) + λ Ω 1 + ... ◮ Insert in the auxiliary equations 1 a 2 − h 2 p ′ ( h ) � � − Π 1 = ∂ x u + O ( λ ) h � u ∂ h Q s − p ′ ( h ) � − Ω 1 = ∂ u Q s ∂ x h + ( − h ∂ h Q s + u ∂ u Q s ) ∂ x u h � b 2 � h 2 − u 2 − g ∂ u Q s + ∂ x z + O ( λ ) ◮ Insert in the physical equations E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Relaxation model Relaxation framework for SW-Exner model Relaxation scheme Towards new models (and other perspectives) Numerical results Stability of the relaxation model ◮ Diffusive physical system W = ( h , u , Z ) T ∂ t W + A ( W ) ∂ x ( W ) = λ∂ x ( D ( W ) ∂ x W ) + O ( λ 2 ) ◮ Diffusion matrix 0 0 0 a 2 − h 2 p ′ ( h ) 1 � � 0 0 D ( W ) = h � � h 2 − u 2 − g ∂ u Q s b 2 × × ◮ Stability requirement b 2 > ( hu ) 2 + gh 2 ∂ u Q s a 2 > h 2 p ′ ( h ) , E. Audusse A relaxation framework for morphodynamics modelling
From SW flows to Morphodynamics Relaxation model Relaxation framework for SW-Exner model Relaxation scheme Towards new models (and other perspectives) Numerical results Relaxation scheme with time splitting ◮ Start from ( H n , u n , Z n ) ◮ Computation of auxiliary variables � Π n � � � gH 2 p ( h n ) 1 � � � ∂ t Π = − Π = λ 2 ⇔ Ω n Q s ( h n , u n ) 1 ∂ t Ω = λ ( Q s − Ω) ” λ = 0” ◮ Solution of homogeneous Riemann problems X R = X ( X l , X r , x , t ) H n +1 , u n +1 , Z n +1 � ◮ Computation of new physical variables � E. Audusse A relaxation framework for morphodynamics modelling
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