How to treat the coupling issue of the Saint-Venant-Exner system of equations Philippe Ung 1 , 4 joint work with Emmanuel Audusse 1 , 2 , Christophe Chalons 3 1 Team ANGE – CEREMA, Inria Rocquencourt, LJLL 2 LAGA – Universit´ e Paris XIII 3 LMV – Universit´ e de Versailles Saint-Quentin-en-Yvelines 4 MAPMO – Universit´ e d’Orl´ eans Egrin June 1 st , 2015
Context & Motivations Numerical scheme Test cases Discussion Outline Context & Motivations Numerical scheme Test cases Discussion Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 2 / 33
Context & Motivations Numerical scheme Test cases Discussion Context & Motivations Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 3 / 33
Context & Motivations Numerical scheme Test cases Discussion Motivations Framework Sediments transport is responsible of modification of river beds. 2 processes of sediments transport: by suspension: particles can be found on the whole vertical water depth and rarely be in contact with the bed, by bedload: particles are moving near the bed by saltation and rolling. Figure: Processes of sediment transport. Thereafter, we only focuse on the bedload transport . Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 4 / 33
Context & Motivations Numerical scheme Test cases Discussion Saint-Venant–Exner equations The model In the literature, most of industrial codes use the Saint-Venant–Exner model . ∂ t H + ∂ x ( Q ) = 0 , (1a) � Q 2 H + gH 2 � = − gH ∂ x B − τ ∂ t Q + ∂ x ρ. (1b) 2 ∂ t B + ∂ x Q s = 0 , (1c) Coupled model between: the Saint-Venant equations (aka shallow-water equations): (1a)–(1b) z H ( t , x ) H ( t , x ): water height, Q ( t , x ) = HU : discharge, U ( t , x ) B ( t , x ): bottom topography, B ( t , x ) with x ∈ Ω ⊆ R , t � 0. x 0 Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 5 / 33
Context & Motivations Numerical scheme Test cases Discussion Saint-Venant–Exner equations The model τ is defined by the Manning formula , Q | Q | τ = ρ gH , (2) s R 4 / 3 H 2 K 2 h where, in the particular case of a rectangular channel with width l , the hydraulic radius R h reads lH R h = l + 2 H . Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 6 / 33
Context & Motivations Numerical scheme Test cases Discussion Saint-Venant–Exner equations The model the Exner equation (1c) where Q s ( t , x ) is the solid transport flux defined by � c ) τ ⋆ g ( ρ s − ρ ) d 3 Q ⋆ s ( τ ⋆ ; τ ⋆ Q s = (3) | τ ⋆ | ρ and the Meyer-Peter-M¨ uller formula , c ) 3 / 2 Q ⋆ s = A ( | τ ⋆ | − τ ⋆ (4) + A a constant, ρ s , ρ resp. the mass densities of the solid and fluid phases, g the gravitational acceleration, with τ ⋆ the shear stress (aka Shields parameter), τ ⋆ the critical value for the initiation of motion, c d the grain diameter. Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 7 / 33
Context & Motivations Numerical scheme Test cases Discussion Saint-Venant–Exner equations The model A more practical expression of the solid discharge Grass formula, Q s = A g U | U | m − 1 (5) where A g is an empirically determined constant and 0 < m < 4. Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 8 / 33
Context & Motivations Numerical scheme Test cases Discussion Saint-Venant–Exner equations The model The Saint-Venant–Exner equations can be rewritten in a vectorial form, ∂ t ˜ W + ∂ x F ( ˜ W ) = S ( ˜ W ) , (6) where HU H HU 2 + gH 2 ˜ F ( ˜ W = HU , W ) = , 2 B Q s 0 S ( ˜ W ) = − gH ∂ x B . 0 Quasilinear form : ∂ t ˜ W + A ( ˜ W ) ∂ x ˜ W = S ( ˜ W ) , where A is the jacobian matrix of F . Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 9 / 33
Context & Motivations Numerical scheme Test cases Discussion Motivations Numerical aspect Two strategies to approximate the solution of the system: splitting and non-splitting methods . Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 10 / 33
Context & Motivations Numerical scheme Test cases Discussion Motivations Numerical aspect Two strategies to approximate the solution of the system: splitting and non-splitting methods . The problem of choice between these two methods remains when considering “fast flow” (Hudson et al, 2003 & 2005): Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 10 / 33
Context & Motivations Numerical scheme Test cases Discussion Motivations Numerical aspect Two strategies to approximate the solution of the system: splitting and non-splitting methods . The problem of choice between these two methods remains when considering “fast flow” (Hudson et al, 2003 & 2005): the splitting method injects numerical instabilities, Figure: Free surface (top) and Bottom topography (bottom) Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 10 / 33
Context & Motivations Numerical scheme Test cases Discussion Motivations Numerical aspect the non-splitting method allows to correct these instabilities, - Roe-type solver (Hudson et al. 2003 & 2005, Murillo and Garcia-Navarro 2010), - Intermediate Field Capturing Riemann solver (Pares 2006, Pares et al. 2011), - Relaxation scheme (Delis et al. 2008, ABCDGJSGS 2011), - Non Homogeneous Riemann solver (Benkhaldoun et al. 2009), - Godunov-type method based on a three-waves Approximate Riemann Solver (ARS). Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 11 / 33
Context & Motivations Numerical scheme Test cases Discussion Numerical scheme Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 12 / 33
Context & Motivations Numerical scheme Test cases Discussion Numerical approximation Properties & Main definitions Positivity of water height, H � 0 , Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 13 / 33
Context & Motivations Numerical scheme Test cases Discussion Numerical approximation Properties & Main definitions Positivity of water height, H � 0 , Well-balanced property or ability to preserve steady states of the lake at rest, U = 0 , H + B = Cte . Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 13 / 33
Context & Motivations Numerical scheme Test cases Discussion Numerical approximation Properties & Main definitions Positivity of water height, H � 0 , Well-balanced property or ability to preserve steady states of the lake at rest, U = 0 , H + B = Cte . Froude number , | U | F r = √ gH . (7) F r < 1 Fluvial regime, F r = 1 Transcritical regime, F r > 1 Torrential regime. Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 13 / 33
Context & Motivations Numerical scheme Test cases Discussion Numerical scheme Objective Main objective Developping a non-splitted method to solve the Saint-Venant–Exner system. Strategy Propose a Godunov-type method to solve the Saint-Venant–Exner equations based on the design of a three-wave Approximate Riemann Solver which is able to degenerate to an ARS satisfying all these properties together when the solid flux is null, sufficiently easy to compute . Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 14 / 33
Context & Motivations Numerical scheme Test cases Discussion Numerical scheme Discretization Space discretization Ω, ∀ i ∈ Z C i x x i − 1 / 2 x i x i +1 / 2 ∆ x i = x i +1 / 2 − x i − 1 / 2 N x : Number of cells. Time discretization t � 0, ∀ n ∈ N t n +1 = t n + ∆ t n , ∆ t > 0 . In the following, we denote ∆ t n = ∆ t . ∆ x i = ∆ x , Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 15 / 33
Context & Motivations Numerical scheme Test cases Discussion Numerical scheme Main ideas Notations: ∀ X ∈ { H , HU , B } , � 0 � ∆ x 1 1 1 � X L ≈ X ( x ) dx ; X R ≈ X ( x ) dx ; X i ≈ X ( x ) dx . ∆ x ∆ x ∆ x − ∆ x 0 C i Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 16 / 33
Context & Motivations Numerical scheme Test cases Discussion Numerical scheme Main ideas Notations: ∀ X ∈ { H , HU , B } , � 0 � ∆ x 1 1 1 � X L ≈ X ( x ) dx ; X R ≈ X ( x ) dx ; X i ≈ X ( x ) dx . ∆ x ∆ x ∆ x − ∆ x 0 C i i ) T a given piecewise At t n , ˜ W n i = ( W n i , B n i ) = ( H n i , H n i U n i , B n constant approximate solution, Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 16 / 33
Context & Motivations Numerical scheme Test cases Discussion Numerical scheme Main ideas Notations: ∀ X ∈ { H , HU , B } , � 0 � ∆ x 1 1 1 � X L ≈ X ( x ) dx ; X R ≈ X ( x ) dx ; X i ≈ X ( x ) dx . ∆ x ∆ x ∆ x − ∆ x 0 C i i ) T a given piecewise At t n , ˜ W n i = ( W n i , B n i ) = ( H n i , H n i U n i , B n constant approximate solution, Building an approximate solution of the Riemann problem at each interface x i +1 / 2 , Egrin – Jun. 1 st , 2015 Ph. Ung Coupling for Sedim. Transp. 16 / 33
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