EGRIN Ecoulements Gravitaires et RIsques Naturels Modeling and Numerics for (Shallow) (Water) Flows E. Audusse . LAGA, UMR 7569, Univ. Paris 13 . GdR EGRIN http ://gdr-egrin.math.cnrs.fr/ May 29, 2017 E. Audusse Numerics for SW flows
EGRIN : Teams ◮ (Applied) Mathematics Paris-Est, Paris-Nord, Paris-Sud, UPMC, Dauphine, Descartes, Nantes, Clermont, Besan¸ con, Montpeliier, Rennes, Bordeuax, Lyon, Nice, Chambery, Orl´ eans, Toulouse, Amiens, Vannes, Toulon, Grenoble, Marseille, Corse, Versailles, Seville ◮ Physics & Mechanics IPGP, IPR, LISAH, IMFT, LTHE, LGGE, ISTO, LMV, LMD, IUSTI, IJRA, Navier, HSM, ETNA, LOF, LPMC, PMMD ◮ INRIA ANGE, LEMON, AIRSEA, CARDAMOM ◮ State Institutes & Companies CEREMA, CERFACS, BRGM, INRA, EDF, ANTEA, LHSV � ≈ 250 members E. Audusse Numerics for SW flows
EGRIN : Board ◮ C. Lucas (MAPMO, Orl´ eans) ◮ J. Sainte-Marie (CEREMA, ANGE Team) ◮ C. Berthon (LJL, Nantes) ◮ F. Bouchut (LAMA, Paris Est) ◮ L. Chupin (LM, Clermont) ◮ S. Cordier (MAPMO, Orl´ eans) ◮ A. Mangeney (IPGP, Paris) ◮ P. Saramito (LJK, Grenoble) ◮ A. Valance (IPR, Rennes & GdR TransNat) ◮ S. Da Veiga (SAFRAN & GdR MascotNum) E. Audusse Numerics for SW flows
EGRIN : Collaborations ◮ GdR TransNat http://transnat.univ-rennes1.fr/ ◮ GdR MePhy M´ ecanique et Physique des Syst` emes Complexes https://www.pmmh.espci.fr/ mephy/wiki/doku.php?id=start ◮ GdR Films Ruissellement et films cisaill´ es https://www.pmmh.espci.fr/ mephy/wiki/doku.php?id=start ◮ GdR Ma-Nu Math´ ematiques pour le Nucl´ eaire http://gdr-manu.math.cnrs.fr/ ◮ GdR Mascot-Num Analyse Stochastique pour Codes et Traitements Num´ eriques http://www.gdr-mascotnum.fr/ E. Audusse Numerics for SW flows
EGRIN : Annual Workshops ◮ Orl´ eans 2013 & 2014 ◮ R. Delannay (Rennes) : Granular flows ◮ J. Garnier (Diderot) : Rare events simulations ◮ A. Valance (Rennes) : Dune dynamics ◮ E. Blayo (INRIA Grenoble) : Data assimilation ◮ E. Fernandez Nieto (S´ eville) : High Order Finite Volumes ◮ P.Y. Lagr´ ee (UPMC) : Hydrodynamics & Erosion models ◮ Nantes 2015 & 2016 ◮ D. G´ erard Varet (Diderot) : Rough boundaries effects ◮ N. Mangold (Nantes) : Gravity driven flows on planets ◮ P. Saramito (Grenoble) : Numerics for Viscoplastic fluids ◮ A.L. Dalibard (UPMC) : Primitive equations of the ocean ◮ T. Lelievre (CERMICS) : Model Reduction Technics ◮ O. Roche (Clermont) : Fluidized Granular Flows ◮ Cargese 2017 ◮ P. Bonneton (Bordeaux) : Dispersive waves ◮ F. Bouchut (UMLV) : Numerical methods for complex rheology ◮ A. Mangeney (IPGP) : Granular geophysical flows E. Audusse Numerics for SW flows
EGRIN : Thematics Le programme scientifique de EGRIN est centr´ e sur la formulation et la r´ esolution num´ erique de mod` eles de complexit´ e r´ eduite par rapport aux ´ equations de Navier-Stokes ` a surface libre, mais s’affranchissant des hypoth` eses trop restricitives qu’on retrouve dans les mod` eles classiques d’´ ecoulements peu profonds. [...] On s’int´ eresse aux ´ ecoulements complexes et aux couplages induits lorsque le fluide interagit avec les sols ou les structures (´ erosion, glissements de terrains...). Les fluides consid´ er´ es sont eux-mˆ emes complexes, au sens o` u ils poss` edent une rh´ eologie particuli` ere (avalanches, ´ ecoulements pyroclastiques...). [...] Sur ces sujets, il est n´ ecessaire de d´ ecloisonner les disciplines et les math´ ematiciens doivent tisser des liens avec des mod´ elisateurs non-math´ ematiciens pour mieux prendre en compte les probl` emes. E. Audusse Numerics for SW flows
EGRIN : Thematics ◮ Hydrodynamics � Tsunamis, Flooding, Dam breaks, Rogue waves... ◮ Complex Fluids � Avalanches, Mud or Pyroclastic flows, Multiphase flows... ◮ Coupling � Morphodynamics, Biological phenomena, Fluid-Structure... ◮ Modeling � Conservation, Energy inequality, Well-posedness... ◮ Numerical Analysis � Accuracy, Robustness, Non linear Stability, WB... ◮ Data Assimilation � Paramester estimation, Filtering, Control... E. Audusse Numerics for SW flows
EGRIN : Basics ◮ (Natural ?) hazards ◮ Simulations with Shallow Water Flows E. Audusse Numerics for SW flows
EGRIN : Basics ◮ Comparisons with Experiments & Measurements E. Audusse Numerics for SW flows
EGRIN : Main Focuses ◮ Hydrostatic NS equations : Multilayer models (ABPS [11], ABPSM [11], Rambaud [12]...) ◮ Non Hydrostatic SW Models : Boussinesq type models (Bonneton et al. [11], Sainte-Marie [11]...) E. Audusse Numerics for SW flows
EGRIN : Main Focuses ◮ Complex flows : Generalized topography, Alternative rheology (Mangeney et al. [03], Bouchut-Westdickenberg [04], Chupin [09], Nieto et al. [10]...) ◮ Erosion processes, Sediment transport (Castro et al. [08], Bouharguane-Mohammadi [09], Cordier et al. [11], ABCDGGJSS [11]...) E. Audusse Numerics for SW flows
EGRIN : Ongoing Work ◮ Hydrostatic NS equations : Multilayer models (Parisot - Vila [14], Di Martino - Haspot [17], Couderc - Duran - Vila [17], AAGP [17]...) ◮ Non Hydrostatic SW Models : Boussinesq type models (Duran - Marche [15], Lannes - Marche [15], Bristeau et al. [15], Aissiouene [16]...) � Non Hydrostatic NS Equations : Layerwise models (Fernandez Nieto - Parisot - Penel - Sainte Marie [17]) E. Audusse Numerics for SW flows
EGRIN : Ongoing Work ◮ Complex flows (Gueugneau-Chupin et al. [17], Bouchut - Mangeney et al. [16], Saramito - Wachs [17], Bristeau et al. [17], Fernandez - Gallardo - Vigneaux [17]...) ◮ Erosion processes, Sediment transport (Fernandez - Narbona et al. [16], Mohamadi [16], Nouhou Bako et al. [17], ABP [17]) E. Audusse Numerics for SW flows
Free Surface Incompressible Navier Stokes Equations ◮ Computational domain z ∈ [ b ( x ) , H ( t , x )] ◮ Equations ∇ · u = 0 , ∂ t u + ( u · ∇ ) u + ∇ p = g + ∇ · σ v ∂ z p = − g + ∇ v · σ v ◮ Boundary conditions E. Audusse Numerics for SW flows
Shallow Water Equations ◮ 1d shallow water equations ∂ t h + ∂ x hu = 0 , hu 2 + gh 2 / 2 � � ∂ t hu + ∂ x = − gh ∂ x b + S f with h : water depth, b : bottom topography, u : velocity of the water column, S f : friction term Saint-Venant (1871), ”Th´ eorie du mouvement non-permanent des eaux, avec application aux crues des rivi` eres et ` a l’introduction des mar´ ees dans leur lit”, Comptes Rendus Acad. Sciences. E. Audusse Numerics for SW flows
Shallow Water Equations : Derivation (SV1) ◮ Derivation ◮ Mass budget E. Audusse Numerics for SW flows
Shallow Water Equations : Derivation (SV1) ◮ Gravity tem ◮ Pressure term ◮ Friction term ◮ Momentum budget E. Audusse Numerics for SW flows
Shallow Water Equations : Derivation (SV1) ◮ Hypothesis : Almost flat bottom ◮ Hypothesis : Constant velocity ◮ Hypothesis : Rectangular channel ◮ Conclusion E. Audusse Numerics for SW flows
Shallow Water Equations : Properties ◮ 2d shallow water equations with sources ∂ t h + ∇ · ( h ¯ u ) = 0 , u + gh 2 � � ∂ t ( h ¯ u ) + ∇ · h ¯ u ⊗ ¯ 2 I = − gh ∇ b − 2Ω × h ¯ u − κ ( h , ¯ u ) ¯ u ◮ Properties ◮ Conservation law ◮ Hyperbolic system (wave propagation, weak solution) ◮ Positivity of water depth (invariant domain, dry zones) ◮ Energy (entropy) (in)equality ◮ Non-trivial steady states (lake at rest) E. Audusse Numerics for SW flows
Shallow Water Equations : Numerics (Finite Volumes) ◮ First Works ◮ Bermudez-Vazquez [’94], Greenberg-Leroux [’96] ◮ Goutal-Maurel [’97] ◮ Positive and well-balanced numerical schemes [’01 → ’16] ◮ Extended Godunov scheme (Chinnayya-Leroux-Seguin) ◮ Kinetic interpretation of source term (Perth.-Sim.,ABBSM) ◮ Extended Suliciu relaxation scheme v1 (Bouchut,Galice,ACU) ◮ Hydrostatic reconstruction (ABBKP, Liang-Marche) ◮ Path-conservative scheme (Castro-Macias-Pares) ◮ Hydrostatic upwind scheme (Berthon-Foucher) ◮ Central scheme (Kurganov, Kurganov-Noelle) ◮ Review books ◮ Hyperbolic Problems & Finite Volumes Method Godlewski-Raviart [’96], Toro [’99], Leveque [’02]... ◮ Bouchut [’04] Nonlinear Stability of FV Methods for Hyperbolic Conservations Laws and WB Schemes for Sources, Birkh¨ auser. E. Audusse Numerics for SW flows
Shallow Water Equations : Numerics (Finite Volumes) ◮ First Works ◮ Bermudez-Vazquez [’94], Greenberg-Leroux [’96] ◮ Goutal-Maurel [’97] ◮ Positive and well-balanced numerical schemes [’01 → ’16] ◮ Extended Godunov scheme (Chinnayya-Leroux-Seguin) ◮ Kinetic interpretation of source term (Perth.-Sim.,ABBSM) ◮ Extended Suliciu relaxation scheme v1 (Bouchut,Galice,ACU) ◮ Hydrostatic reconstruction (ABBKP, Liang-Marche) ◮ Path-conservative scheme (Castro-Macias-Pares) ◮ Hydrostatic upwind scheme (Berthon-Foucher) ◮ Central scheme (Kurganov, Kurganov-Noelle) ◮ Review books ◮ Hyperbolic Problems & Finite Volumes Method Godlewski-Raviart [’96], Toro [’99], Leveque [’02]... ◮ Bouchut [’04] Nonlinear Stability of FV Methods for Hyperbolic Conservations Laws and WB Schemes for Sources, Birkh¨ auser. E. Audusse Numerics for SW flows
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