Generalised Serre-Green-Naghdi equations for open channel and for - PowerPoint PPT Presentation

Geometric set-up & Equations Incompressible and irrotational Euler equations div( ρ 0 u ) = 0 , ∂ ∂t ( ρ 0 u ) + div( ρ 0 u ⊗ u ) + ∇ p − ρ 0 F = 0 with u = ( u, v, w ) : velocity field ρ 0 : density F = (0 , 0 , − g ) : external force p : pressure M. Ersoy (IMATH) 3D-1D 2020, 20 October 5 / 21

Geometric set-up & Equations Incompressible and irrotational Euler equations div( ρ 0 u ) = 0 , ∂ ∂t ( ρ 0 u ) + div( ρ 0 u ⊗ u ) + ∇ p − ρ 0 F = 0 with completed with the irrotational relations u = ( u, v, w ) : velocity field ∂y = ∂v ∂u ∂x, ∂v ∂z = ∂w ∂y , ∂u ∂z = ∂w ρ 0 : density ∂x . F = (0 , 0 , − g ) : external force p : pressure M. Ersoy (IMATH) 3D-1D 2020, 20 October 5 / 21

Geometric set-up & Equations Incompressible and irrotational Euler equations div( ρ 0 u ) = 0 , ∂ ∂t ( ρ 0 u ) + div( ρ 0 u ⊗ u ) + ∇ p − ρ 0 F = 0 with completed with the irrotational relations u = ( u, v, w ) : velocity field ∂y = ∂v ∂u ∂x, ∂v ∂z = ∂w ∂y , ∂u ∂z = ∂w ρ 0 : density ∂x . F = (0 , 0 , − g ) : external force p : pressure M. Ersoy (IMATH) 3D-1D 2020, 20 October 5 / 21

Geometric set-up & Equations Incompressible and irrotational Euler equations div( ρ 0 u ) = 0 , ∂ ∂t ( ρ 0 u ) + div( ρ 0 u ⊗ u ) + ∇ p − ρ 0 F = 0 free surface kinematic boundary condition, u · n fs = ∂ ∂t m · n fs and p = p 0 , ∀ m ( t, x, y ) = ( x, y, η ( t, x, y )) ∈ Γ fs ( t, x ) no-penetration condition on the wet boundary u · n wb = 0 , ∀ m ( x, y ) = ( x, y, d ( x, y )) ∈ Γ wb ( x ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 5 / 21

Rescaling and asymptotic regime Let us define the dispersive parameters µ 1 = h 2 1 L 2 µ 2 = H 2 2 L 2 , such that h 1 < H 1 = H 2 ≪ L, i.e. µ 1 < µ 2 2 where H 1 : characteristic scale of channel width h 1 : characteristic wave-length in the transversal direction H 2 : characteristic water depth U √ gH 2 F r = : Froude’s number T = L : characteristic time U P = U 2 : characteristic pressure X : characteristic length of x M. Ersoy (IMATH) 3D-1D 2020, 20 October 6 / 21

Rescaling and asymptotic regime Then, define the dimensionless variables x = x P = P ϕ = ϕ � � L, P , � , h 1 y = y u = u d = d � � , � U , , h 1 H 2 z = z v = v v η = η , V = √ µ 1 U , . � � � H 2 H 2 t = t w = w w � T , = √ µ 2 U . � W M. Ersoy (IMATH) 3D-1D 2020, 20 October 6 / 21

Rescaling and asymptotic regime We get ∂u ∂x + ∂v ∂y + ∂w ∂z = 0 ∂u ∂t + u∂u ∂x + v ∂u ∂y + w∂u ∂z + ∂P ∂x = 0 � ∂v � ∂t + u∂v ∂x + v ∂v ∂y + w∂v + ∂P µ 1 ∂y = 0 ∂z � ∂w � ∂t + u∂w ∂x + v ∂w ∂y + w∂w + ∂P ∂z = − 1 µ 2 2 ∂z F r and ∂u ∂v ∂v ∂w ∂y , ∂u ∂w ∂y = µ 1 ∂x, µ 1 ∂z = µ 2 ∂z = µ 2 ∂x . M. Ersoy (IMATH) 3D-1D 2020, 20 October 6 / 21

”Coulisses” I : why µ 1 � = µ 2 ? µ 1 = µ 2 ⇒ no analytical expression of the asymptotic terms. M. Ersoy (IMATH) 3D-1D 2020, 20 October 7 / 21

”Coulisses” I : why µ 1 � = µ 2 ? µ 1 = µ 2 ⇒ no analytical expression of the asymptotic terms. Indeed, in 2D-1D reduction, we proceed as follows u x + w z = 0 M. Ersoy (IMATH) 3D-1D 2020, 20 October 7 / 21

”Coulisses” I : why µ 1 � = µ 2 ? µ 1 = µ 2 ⇒ no analytical expression of the asymptotic terms. Indeed, in 2D-1D reduction, we proceed as follows �� z � u x + w z = 0 +BC ⇒ w ( t, x, z ) = − u ( t, x, z ) dz d x M. Ersoy (IMATH) 3D-1D 2020, 20 October 7 / 21

”Coulisses” I : why µ 1 � = µ 2 ? µ 1 = µ 2 ⇒ no analytical expression of the asymptotic terms. Indeed, in 2D-1D reduction, we proceed as follows �� z � u x + w z = 0 +BC ⇒ w ( t, x, z ) = − u ( t, x, z ) dz d x u z = µw x M. Ersoy (IMATH) 3D-1D 2020, 20 October 7 / 21

”Coulisses” I : why µ 1 � = µ 2 ? µ 1 = µ 2 ⇒ no analytical expression of the asymptotic terms. Indeed, in 2D-1D reduction, we proceed as follows �� z � u x + w z = 0 +BC ⇒ w ( t, x, z ) = − u ( t, x, z ) dz d x � z u z = µw x ⇒ u ( t, x, z ) = u | z = d ( t, x ) + µ w x ( t, x, z ) dz d M. Ersoy (IMATH) 3D-1D 2020, 20 October 7 / 21

”Coulisses” I : why µ 1 � = µ 2 ? µ 1 = µ 2 ⇒ no analytical expression of the asymptotic terms. Indeed, in 2D-1D reduction, we proceed as follows �� z � u x + w z = 0 +BC ⇒ w ( t, x, z ) = − u ( t, x, z ) dz d x � z u z = µw x ⇒ u ( t, x, z ) = u | z = d ( t, x ) + µ w x ( t, x, z ) dz ⇒ �� z � d w ( t, x, z ) = − u | z = d ( t, x ) dz + O ( µ ) d x M. Ersoy (IMATH) 3D-1D 2020, 20 October 7 / 21

”Coulisses” I : why µ 1 � = µ 2 ? µ 1 = µ 2 ⇒ no analytical expression of the asymptotic terms. Indeed, in 2D-1D reduction, we proceed as follows �� z � u x + w z = 0 +BC ⇒ w ( t, x, z ) = − u ( t, x, z ) dz d x � z u z = µw x ⇒ u ( t, x, z ) = u | z = d ( t, x ) + µ w x ( t, x, z ) dz ⇒ �� z � d w ( t, x, z ) = − u | z = d ( t, x ) dz + O ( µ ) d x ⇒ u ( t, x, z ) = f 1 ( u | z = d ( t, x )) + µf 2 ( z, u | z = d ( t, x ) , d ( x )) + O ( µ 2 ) ⇒ u | z = d = f 3 (¯ u ( t, x )) . . . M. Ersoy (IMATH) 3D-1D 2020, 20 October 7 / 21

”Coulisses” I : why µ 1 � = µ 2 ? µ 1 = µ 2 ⇒ no analytical expression of the asymptotic terms. Indeed, in 3D-1D reduction, we proceed as follows � u x + v y + w z = 0 ⇒ v y + w z dydz . . . Ω M. Ersoy (IMATH) 3D-1D 2020, 20 October 7 / 21

”Coulisses” I : why µ 1 � = µ 2 ? µ 1 = µ 2 ⇒ no analytical expression of the asymptotic terms. Indeed, in 3D-1D reduction, we proceed as follows � u x + v y + w z = 0 ⇒ v y + w z dydz . . . Ω Therefore, we assume µ 1 � = µ 2 . M. Ersoy (IMATH) 3D-1D 2020, 20 October 7 / 21

”Coulisses” II : why introduce h 1 < H 1 ? A counter example if h 1 = H 1 : Consider the (nondimensional) rectangular channel � � � � 0 , H 1 0 , L c (˜ x, ˜ y, ˜ z ) ∈ × × [0 , 1] where L ≪ L c . L h 1 M. Ersoy (IMATH) 3D-1D 2020, 20 October 8 / 21

”Coulisses” II : why introduce h 1 < H 1 ? A counter example if h 1 = H 1 : Consider the (nondimensional) rectangular channel � � � � 0 , H 1 0 , L c (˜ x, ˜ y, ˜ z ) ∈ × × [0 , 1] where L ≪ L c . L h 1 w ) T = ∇ ˜ Incompressible + Irrotational ⇒ ∃ ˜ φ ; (˜ u, ˜ v, ˜ φ solution of φ + 1 φ + 1 x ˜ y ˜ z ˜ ∂ 2 ∂ 2 ∂ 2 φ = 0 . x ˜ ˜ y ˜ ˜ ˜ z ˜ µ 1 µ 2 M. Ersoy (IMATH) 3D-1D 2020, 20 October 8 / 21

”Coulisses” II : why introduce h 1 < H 1 ? A counter example if h 1 = H 1 : Consider the (nondimensional) rectangular channel � � � � 0 , H 1 0 , L c (˜ x, ˜ y, ˜ z ) ∈ × × [0 , 1] where L ≪ L c . L h 1 w ) T = ∇ ˜ Incompressible + Irrotational ⇒ ∃ ˜ φ ; (˜ u, ˜ v, ˜ φ More precisely, ∀ ( p, q ) ∈ N 2 , we have : � � � � � � � cosh h 2 p 2 µ 2 L 2 c + q 2 µ 2 π ˜ z 1 pπ ˜ xL qπ ˜ yh 1 L 2 µ 1 H 2 ˜ 1 φ p,q ( x, y, z ) = cos cos � � � . L c H 1 h 2 p 2 µ 2 L 2 c + q 2 µ 2 cosh π 1 L 2 H 2 µ 1 1 M. Ersoy (IMATH) 3D-1D 2020, 20 October 8 / 21

”Coulisses” II : why introduce h 1 < H 1 ? A counter example if h 1 = H 1 : Consider the (nondimensional) rectangular channel � � � � 0 , H 1 0 , L c (˜ x, ˜ y, ˜ z ) ∈ × × [0 , 1] where L ≪ L c . L h 1 w ) T = ∇ ˜ Incompressible + Irrotational ⇒ ∃ ˜ φ ; (˜ u, ˜ v, ˜ φ More precisely, ∀ ( p, q ) ∈ N 2 , we have : � � � � � � � cosh h 2 p 2 µ 2 L 2 c + q 2 µ 2 π ˜ z 1 pπ ˜ xL qπ ˜ yh 1 L 2 µ 1 H 2 ˜ 1 φ p,q ( x, y, z ) = cos cos � � � . L c H 1 h 2 p 2 µ 2 L 2 c + q 2 µ 2 cosh π 1 L 2 H 2 µ 1 1 Keeping in mind that H 2 < L ≪ L c , ◮ if h 1 = H 1 < H 2 then p 2 µ 2 L 2 + q 2 µ 2 x ˜ µ 1 ⇒ ˜ u = ∂ ˜ φ is rapidly varying in ˜ z L 2 c unless H 1 > H 2 (out of context) M. Ersoy (IMATH) 3D-1D 2020, 20 October 8 / 21

”Coulisses” II : why introduce h 1 < H 1 ? A counter example if h 1 = H 1 : Consider the (nondimensional) rectangular channel � � � � 0 , H 1 0 , L c (˜ x, ˜ y, ˜ z ) ∈ × × [0 , 1] where L ≪ L c . L h 1 w ) T = ∇ ˜ Incompressible + Irrotational ⇒ ∃ ˜ φ ; (˜ u, ˜ v, ˜ φ More precisely, ∀ ( p, q ) ∈ N 2 , we have : � � � � � � � cosh h 2 p 2 µ 2 L 2 c + q 2 µ 2 π ˜ z 1 pπ ˜ xL qπ ˜ yh 1 L 2 µ 1 H 2 ˜ 1 φ p,q ( x, y, z ) = cos cos � � � . L c H 1 h 2 p 2 µ 2 L 2 c + q 2 µ 2 cosh π 1 L 2 H 2 µ 1 1 Keeping in mind that H 2 < L ≪ L c , ◮ if h 1 = H 1 < H 2 then is rapidly varying in ˜ z ◮ Therefore, we consider h 1 < H 1 = H 2 : p 2 µ 2 L 2 h 2 = p 2 H 2 + q 2 H 2 + q 2 µ 2 1 2 2 L 2 H 2 L 2 H 2 µ 1 c 1 c 1 M. Ersoy (IMATH) 3D-1D 2020, 20 October 8 / 21

”Coulisses” III : order of integration ” Coulisses”II naturally yields to V < W < U where ( U, V = √ µ 1 U, W = √ µ 2 U ) As a consequence, we proceed as follows ◮ 3D-2D reduction (width averaging) ◮ 2D-1D reduction (depth averaging) ◮ 3D-1D reduction (section averaging) M. Ersoy (IMATH) 3D-1D 2020, 20 October 9 / 21

”Coulisses” III : order of integration ” Coulisses”II naturally yields to V < W < U where ( U, V = √ µ 1 U, W = √ µ 2 U ) As a consequence, we proceed as follows ◮ 3D-2D reduction (width averaging) : u ( t, x, y, z ) = � u � ( t, x, z ) + O ( µ 1 ) ◮ 2D-1D reduction (depth averaging) ◮ 3D-1D reduction (section averaging) M. Ersoy (IMATH) 3D-1D 2020, 20 October 9 / 21

”Coulisses” III : order of integration ” Coulisses”II naturally yields to V < W < U where ( U, V = √ µ 1 U, W = √ µ 2 U ) As a consequence, we proceed as follows ◮ 3D-2D reduction (width averaging) : u ( t, x, y, z ) = � u � ( t, x, z ) + O ( µ 1 ) ◮ 2D-1D reduction (depth averaging) : � u � ( t, x, z ) = u ( t, x ) + µ 2 f ( u ( t, x ) , Ω( t, x )) + O ( µ 2 2 ) where u ( t, x ) is the section-averaged velocity ◮ 3D-1D reduction (section averaging) M. Ersoy (IMATH) 3D-1D 2020, 20 October 9 / 21

”Coulisses” III : order of integration ” Coulisses”II naturally yields to V < W < U where ( U, V = √ µ 1 U, W = √ µ 2 U ) As a consequence, we proceed as follows ◮ 3D-2D reduction (width averaging) : u ( t, x, y, z ) = � u � ( t, x, z ) + O ( µ 1 ) ◮ 2D-1D reduction (depth averaging) : � u � ( t, x, z ) = u ( t, x ) + µ 2 f ( u ( t, x ) , Ω( t, x )) + O ( µ 2 2 ) where u ( t, x ) is the section-averaged velocity ◮ 3D-1D reduction (section averaging) : u ( t, x, y, z ) = u ( t, x ) + µ 2 f ( u ( t, x ) , Ω( t, x )) + O ( µ 2 2 ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 9 / 21

Outline Outline 1 Derivation (based on Euler equations) 3D-2D 2D-1D 3D-1D 2 Improved model and stability Reformulated and stable models Invertible operator 3 Numerical analysis and test case Finite Volume scheme Numerical simulation 4 Conclusion and perspectives M. Ersoy (IMATH) 3D-1D 2020, 20 October 9 / 21

Step 1 : 3D-2D reduction Div and irrotational equations ⇒ noting X α ( t, x, z ) := X ( t, x, α ( x, z ) , z ) we have � µ 2 u α ( t, x, z ) − µ 1 ∂ � w α ( t, x, z )( y − α ( x, z )) 2 � 1 � u ( t, x, y, z ) = ∂x div x,z + O 2 µ 2 and � µ 2 � w α ( t, x, z ) − µ 1 ∂ w α ( t, x, z )( y − α ( x, z )) 2 � 1 � w ( t, x, y, z ) = ∂z div x,z + O µ 2 2 µ 2 2 M. Ersoy (IMATH) 3D-1D 2020, 20 October 10 / 21

Step 1 : 3D-2D reduction Width-averaging ⇒ noting � β ( x,z ) 1 � X � ( t, x, z ) := X ( t, x, y, z ) dy σ ( x, z ) α ( x,z ) we have � µ 2 σ ( x, z ) u α ( t, x, z ) − µ 1 ∂ � w α ( t, x, z ) σ ( x, z ) 3 � 1 � σ ( x, z ) � u � ( t, x, z ) = ∂x div x,z + O , 6 µ 2 � µ 2 � σ ( x, z ) w α ( t, x, z ) − µ 1 ∂ w α ( t, x, z ) σ ( x, z ) 3 � 1 σ ( x, z ) � w � ( t, x, z ) = ∂z div x,z � + O µ 2 6 µ 2 2 where σ ( x, z ) = β ( x, z ) − α ( x, z ) is the width of the section at the elevation z . M. Ersoy (IMATH) 3D-1D 2020, 20 October 10 / 21

Step 1 : 3D-2D reduction Width-averaging ⇒ � η ( t,x,y ) P ( t, x, y, z ) = P α ( t, x, z )+ O ( µ 1 ) = η ( t, x, y ) − z D + µ 2 Dtw α ( t, x, z ) ds + O ( µ 1 ) F r 2 z (a) Initial a . Debyaoui, Ersoy, Asymptotic Analysis, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 10 / 21

Step 1 : 3D-2D reduction Width-averaging ⇒ � η ( t,x,y ) P ( t, x, y, z ) = P α ( t, x, z )+ O ( µ 1 ) = η ( t, x, y ) − z D + µ 2 Dtw α ( t, x, z ) ds + O ( µ 1 ) F r 2 z ⇓ Flat free surface approximation a : η ( t, x, y ) = η eq ( t, x ) + O ( µ 1 ) ⇒ (c) Initial (d) Flat FS approximation a . Debyaoui, Ersoy, Asymptotic Analysis, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 10 / 21

Step 1 : 3D-2D reduction Width-averaging ⇒ we get the 2D width-averaged model � µ 2 � µ 1 ∂ σ ∂ � w α σ 3 ��� � � 1 div x,z [ σ w α ] + O = div x,z µ 2 6 µ 2 ∂z ∂z 2 � � µ 2 ∂t ( σu α ) + div x,z [ σu α w α ] + ∂ ∂ ∂σ 1 ∂x ( σP α ) + O = P α µ 2 ∂x 2 + µ 1 ∂ � ∂ w α σ 3 �� � u α ∂z div x,z 6 µ 2 ∂x � ∂ � + ∂ − σ µ 2 ∂t ( σw α ) + div x,z [ σw α w α ] ∂z ( σP α ) = F r 2 ∂σ + P α ∂z + O ( µ 1 ) completed with the irrotational equation ∂u α ∂w α ∂z = µ 2 ∂x + O ( µ 1 ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 10 / 21

Outline Outline 1 Derivation (based on Euler equations) 3D-2D 2D-1D 3D-1D 2 Improved model and stability Reformulated and stable models Invertible operator 3 Numerical analysis and test case Finite Volume scheme Numerical simulation 4 Conclusion and perspectives M. Ersoy (IMATH) 3D-1D 2020, 20 October 10 / 21

Step 2 : 2D-1D reduction Div and irrotational equations (model 2D) ⇒ noting � z 1 ∂ f b ( t, x ) = f α ( t, x, d ∗ ( x )) , S ( u, x, z ) = ∂x ( uS ( x, z )) , S ( x, z ) = σ ( x, s ) ds σ ( x, z ) d ∗ ( x ) we have � z ∂ ∂x S ( u b , x, s ) ds + O ( µ 2 u α ( t, x, z ) = u b ( t, x ) − µ 2 2 ) d ∗ ( x ) and 1 ∂ w α ( t, x, z ) = − ∂x ( u b ( t, x ) S ( x, z )) + O ( µ 2 ) σ ( x, z ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 11 / 21

Step 2 : 2D-1D reduction Depth-averaging ⇒ noting � η eq ( t,x ) � β ( x,z ) 1 ¯ u eq = u ( t, x, y, z ) dydz A eq ( t, x ) d ∗ ( x ) α ( x,z ) we get u b ( t, x ) = u eq ( t, x ) ¯ �� z � � η eq ( t,x ) µ 2 ∂ + σ ( x, z ) ∂x S (¯ u eq ( t, x ) , x, s ) ds dz A eq ( t, x ) d ∗ ( x ) d ∗ ( x ) + O ( µ 2 2 ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 11 / 21

Step 2 : 2D-1D reduction Depth-averaging ⇒ finally, u eq , x, z ) + O ( µ 2 u ( t, x, y, z ) = ¯ u eq ( t, x ) + µ 2 B 0 (¯ 2 ) with � � � η eq ( t,x ) � z 1 ∂ B 0 (¯ u eq , x, z ) = σ ( x, z ) ∂x S (¯ u eq ( t, x ) , x, s ) ds dz A eq ( t, x ) d ∗ ( x ) d ∗ ( x ) � z ∂ − ∂x S (¯ u eq ( t, x ) , x, s ) ds d ∗ ( x ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 11 / 21

Step 2 : 2D-1D reduction Depth-averaging ⇒ we also have P ( t, x, y, z ) = P h ( t, x, z ) + µ 2 P nh ( t, x, z ) + O ( µ 2 2 ) where P h ( t, x, z ) = ( z − η eq ( t, x )) 2 F r and � η eq ( t,x ) � u eq ( t, x ) , x, s )) 2 � 1 ∂ P nh ( t, x, z ) = ( σ ( x, s ) S (¯ ds 2 σ ( x, s ) 2 ∂z z � η eq ( t,x ) ∂ − ∂t S (¯ u eq ( t, x ) , x, s ) z + ¯ u eq ( t, x ) ∂ ∂x ( σ ( x, s ) S (¯ u eq ( t, x ) , x, s )) ds σ ( x, s ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 11 / 21

Outline Outline 1 Derivation (based on Euler equations) 3D-2D 2D-1D 3D-1D 2 Improved model and stability Reformulated and stable models Invertible operator 3 Numerical analysis and test case Finite Volume scheme Numerical simulation 4 Conclusion and perspectives M. Ersoy (IMATH) 3D-1D 2020, 20 October 11 / 21

Step 3 : 3D-1D reduction Euler equations in Ω eq instead of Ω M. Ersoy (IMATH) 3D-1D 2020, 20 October 12 / 21

Step 3 : 3D-1D reduction Euler equations in Ω eq instead of Ω Boundary condition : � ∂ � � ∂t M + u ∂ ∂x M − v · n ds = 0 ∂ Ω eq ( t,x ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 12 / 21

Step 3 : 3D-1D reduction Euler equations in Ω eq instead of Ω Boundary condition : � ∂ � � ∂t M + u ∂ ∂x M − v · n ds = 0 ∂ Ω eq ( t,x ) Introduce wet region indicator function Φ which satisfies � ∂t Φ + ∂ ∂ ∂x (Φ u ) + div y,z [Φ v ] = 0 on Ω eq ( t ) = Ω eq ( t, x ) . 0 ≤ x ≤ 1 where v = ( v, w ) . M. Ersoy (IMATH) 3D-1D 2020, 20 October 12 / 21

Step 3 : 3D-1D reduction Euler equations in Ω eq instead of Ω Boundary condition : � ∂ � � ∂t M + u ∂ ∂x M − v · n ds = 0 ∂ Ω eq ( t,x ) Introduce wet region indicator function Φ which satisfies � ∂t Φ + ∂ ∂ ∂x (Φ u ) + div y,z [Φ v ] = 0 on Ω eq ( t ) = Ω eq ( t, x ) . 0 ≤ x ≤ 1 where v = ( v, w ) . Section-averaging equations using the approximation u eq , x, z ) + O ( µ 2 u ( t, x, y, z ) = u eq ( t, x ) + µ 2 B 0 (¯ ¯ 2 ) η ( t, x, y ) = η eq ( t, x ) + O ( µ 1 ) P h ( t, x, z ) + µ 2 P nh ( t, x, z ) + O ( µ 2 P ( t, x, y, z ) = 2 ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 12 / 21

The new 1D nonlinear and dispersive model  ∂tA eq + ∂ ∂   ∂xQ eq = 0   � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( DI 1 ( x, A eq , Q eq )) =   ∂x A eq   I 2 ( x, A eq ) + µ 2 DI 2 ( x, A eq , Q eq ) + O ( µ 2 2 ) where � A eq = dy dz : wet area Ω eq ( t,x ) Q eq = A eq ( t, x )¯ u eq ( t, x ) : discharge ◮ Debyaoui, Ersoy. Asymptotic Analysis, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 13 / 21

The new 1D nonlinear and dispersive model  ∂tA eq + ∂ ∂   ∂xQ eq = 0   � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( DI 1 ( x, A eq , Q eq )) =   ∂x A eq   I 2 ( x, A eq ) + µ 2 DI 2 ( x, A eq , Q eq ) + O ( µ 2 2 ) where � η eq ( t, x ) − z I 1 = σ ( x, z ) dy dz : hydro. press. F 2 Ω eq ( t,x ) r � y + ( t,x ) h eq ( t, x ) ∂ I 2 = − ∂xd ( x, y ) dy : hydro. press. source F r 2 y − ( t,x ) ◮ Debyaoui, Ersoy. Asymptotic Analysis, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 13 / 21

The new 1D nonlinear and dispersive model  ∂tA eq + ∂ ∂   ∂xQ eq = 0   � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( DI 1 ( x, A eq , Q eq )) =   ∂x A eq   I 2 ( x, A eq ) + µ 2 DI 2 ( x, A eq , Q eq ) + O ( µ 2 2 ) where � DI 1 = P nh ( t, x, z ) dy dz : (disp) non hydro. press. Ω eq ( t,x ) � y + ( t,x ) P nh ( t, x, d ( x, y )) ∂ DI 2 = − ∂xd ( x, y ) dy : (disp) non hydro. press. source y − ( t,x ) ◮ Debyaoui, Ersoy. Asymptotic Analysis, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 13 / 21

The new 1D nonlinear and dispersive model  ∂tA eq + ∂ ∂   ∂xQ eq = 0   � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( DI 1 ( x, A eq , Q eq )) =   ∂x A eq   I 2 ( x, A eq ) + µ 2 DI 2 ( x, A eq , Q eq ) + O ( µ 2 2 ) Remark (Generalisation of the free surface model) Setting µ 2 = 0 , we recover the usual nlsw equations for open channel. ◮ Bourdarias, Ersoy, Gerbi. Science China Mathematics, 2012. ◮ Debyaoui, Ersoy. Asymptotic Analysis, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 13 / 21

Reformulation : generalization of the SGN equations  ∂tA eq + ∂ ∂  ∂xQ eq = 0    � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( D (¯ u eq ) G ( A eq , x )) = I 2 ( x, A eq )   ∂x A eq   u eq , S, σ ) + O ( µ 2 + µ 2 G (¯ 2 ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 14 / 21

Reformulation : generalization of the SGN equations  ∂tA eq + ∂ ∂  ∂xQ eq = 0    � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( D (¯ u eq ) G ( A eq , x )) = I 2 ( x, A eq )   ∂x A eq   u eq , S, σ ) + O ( µ 2 + µ 2 G (¯ 2 ) where � ∂ � 2 − ∂ ∂ ∂ ∂ D (¯ u eq ) = ∂x ¯ u eq ∂x ¯ u eq − ¯ u eq ∂x ¯ u eq ∂t ∂x and � η eq � η eq S ( x, s ) G ( A eq , x ) = σ ( x, z ) σ ( x, s ) ds dz d ∗ ( x ) z M. Ersoy (IMATH) 3D-1D 2020, 20 October 14 / 21

Reformulation : generalization of the SGN equations  ∂tA eq + ∂ ∂  ∂xQ eq = 0    � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( D (¯ u eq ) G ( A eq , x )) = I 2 ( x, A eq )   ∂x A eq   u eq , S, σ ) + O ( µ 2 + µ 2 G (¯ 2 ) where   ∂xS ( x, s ) ∂ ∂ � η eq ∂xσ ( x, s ) u 2 − ∂ ∂   G ( u, S, σ ) = ∂xS ( x, s )   σ ( x, s ) σ ( x, s ) ∂x z � S ( x, s ) ∂ � u 2 ∂xσ ( x, s ) . + ∂ σ ( x, s ) 2 ∂x 2 � ∂ � ∂ ∂xS ( x, s ) ∂ − ∂t ¯ u eq + ¯ u eq ∂x ¯ u eq ds σ ( x, s ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 14 / 21

Reformulation : generalization of the SGN equations  ∂tA eq + ∂ ∂  ∂xQ eq = 0    � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( D (¯ u eq ) G ( A eq , x )) = I 2 ( x, A eq )   ∂x A eq   u eq , S, σ ) + O ( µ 2 + µ 2 G (¯ 2 ) Setting σ = 1 , d = 1 , A eq = h eq S ( x, z ) ≡ S ( z ) ⇒ G = 0 and I 2 = 0 3 G = h eq 3 2 I 1 = h eq 2 F 2 r M. Ersoy (IMATH) 3D-1D 2020, 20 October 14 / 21

Reformulation : generalization of the SGN equations  ∂tA eq + ∂ ∂  ∂xQ eq = 0    � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( D (¯ u eq ) G ( A eq , x )) = I 2 ( x, A eq )   ∂x A eq   u eq , S, σ ) + O ( µ 2 + µ 2 G (¯ 2 ) we recover the classical SGN equations on flat bottom  ∂th eq + ∂ ∂   ∂x ( h eq u eq ) = 0 � � � h eq � 2 3 ∂t ( h eq u eq ) + ∂ ∂ 2 + h eq ∂  = O ( µ 2  h eq u eq + µ 2 3 D ( u eq ) 2 ) 2 F 2 ∂x ∂x r M. Ersoy (IMATH) 3D-1D 2020, 20 October 14 / 21

Reformulation : generalization of the SGN equations  ∂tA eq + ∂ ∂  ∂xQ eq = 0    � Q eq � 2 ∂tQ eq + ∂ ∂ ∂ + I 1 ( x, A eq ) + µ 2 ∂x ( D (¯ u eq ) G ( A eq , x )) = I 2 ( x, A eq )   ∂x A eq   u eq , S, σ ) + O ( µ 2 + µ 2 G (¯ 2 ) Remark Dispersive equation are usually characterized by third order term ⇒ may create high frequencies instabilities Figure – Bourdarias, Gerbi, and Ralph Lteif. Computers & Fluids, 156 :283–304, 2017. M. Ersoy (IMATH) 3D-1D 2020, 20 October 14 / 21

Outline Outline 1 Derivation (based on Euler equations) 3D-2D 2D-1D 3D-1D 2 Improved model and stability Reformulated and stable models Invertible operator 3 Numerical analysis and test case Finite Volume scheme Numerical simulation 4 Conclusion and perspectives M. Ersoy (IMATH) 3D-1D 2020, 20 October 14 / 21

Outline Outline 1 Derivation (based on Euler equations) 3D-2D 2D-1D 3D-1D 2 Improved model and stability Reformulated and stable models Invertible operator 3 Numerical analysis and test case Finite Volume scheme Numerical simulation 4 Conclusion and perspectives M. Ersoy (IMATH) 3D-1D 2020, 20 October 14 / 21

A more stable formulation → useful for numerical purpose Define the linear T and the quadratic Q operators � η eq � η eq T [ A eq , d, σ, z ]( u ) = ∂ S ( x, s ) 1 ∂ ∂x ( u ) σ ( x, s ) ds + u ∂xS ( x, s ) ds , σ ( x, s ) z z and � ∂ � 2 S ( x, s ) � η eq G [ A eq , d, σ, z ]( u ) = 2 ∂xu σ ( x, s ) + z   ∂xS ( x, s ) ∂ ∂ ∂xσ ( x, s ) u 2 − ∂ ∂   ∂xS ( x, s )   σ ( x, s ) σ ( x, s ) ∂x � S ( x, s ) ∂ � u 2 ∂xσ ( x, s ) + ∂ ds ∂x 2 σ ( x, s ) 2 M. Ersoy (IMATH) 3D-1D 2020, 20 October 15 / 21

A more stable formulation → useful for numerical purpose Define the linear T and the quadratic Q operators Define the averaged linear T and the quadratic Q operators � η eq T [ A eq , d, σ ]( u, ψ ) = ψ T [ A eq , d, σ, z ]( u ) dz d ∗ ( x ) and � η eq G [ A eq , d, σ ]( u, ψ ) = ψ G [ A eq , d, σ, z ]( u ) dz d ∗ ( x ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 15 / 21

A more stable formulation → useful for numerical purpose Define the linear T and the quadratic Q operators Define the averaged linear T and the quadratic Q operators Define the operators L and Q � u � L [ A eq , d, σ ]( u ) = A eq L [ A eq , d, σ ] A eq and � ∂ � �� � � 1 u, ∂ Q [ A eq , d, σ ]( u ) = G [ A eq , d, σ ] ( u, σ ) − G [ A eq , d, σ ] ∂xσ A eq ∂x M. Ersoy (IMATH) 3D-1D 2020, 20 October 15 / 21

A more stable formulation → useful for numerical purpose Define the linear T and the quadratic Q operators Define the averaged linear T and the quadratic Q operators Define the operators L and Q and finally the operator L � u � L [ A eq , d, σ ]( u ) = A eq L [ A eq , d, σ ] A eq M. Ersoy (IMATH) 3D-1D 2020, 20 October 15 / 21

A more stable formulation → useful for numerical purpose Define the linear T and the quadratic Q operators Define the averaged linear T and the quadratic Q operators Define the operators L and Q and finally the operator L Reformulated model  ∂tA eq + ∂ ∂  ∂x ( A eq u eq ) = 0   � � ∂ 2 �� ∂t ( A eq u eq ) + ∂ + ∂ � � I d − µ 2 L [ A eq , d, σ ] ∂xI 1 ( x, A eq ) A eq u eq   ∂x  + µ 2 A eq Q [ A eq , d, σ ]( u eq ) = I 2 ( x, A eq ) + O ( µ 2 2 ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 15 / 21

A more stable formulation → useful for numerical purpose Define the linear T and the quadratic Q operators Define the averaged linear T and the quadratic Q operators Define the operators L and Q and finally the operator L Reformulated model  ∂tA eq + ∂ ∂  ∂x ( A eq u eq ) = 0   � � ∂ 2 �� ∂t ( A eq u eq ) + ∂ + ∂ � � I d − µ 2 L [ A eq , d, σ ] ∂xI 1 ( x, A eq ) A eq u eq   ∂x  + µ 2 A eq Q [ A eq , d, σ ]( u eq ) = I 2 ( x, A eq ) + O ( µ 2 2 ) Remark Inverting I d − µ 2 L [ A eq , d, σ ] ⇒ no third order term ⇒ more stable formulation ◮ Bonneton, Barth´ elemy, Chazel, Cienfuegos, Lannes, Marche, and Tissier. European Journal of Mechanics-B/Fluids, 2011 ◮ Debyaoui, Ersoy. Part 2, preprint, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 15 / 21

A more stable formulation → useful for numerical purpose Define the linear T and the quadratic Q operators Define the averaged linear T and the quadratic Q operators Define the operators L and Q and finally the operator L Reformulated model  ∂tA eq + ∂ ∂  ∂x ( A eq u eq ) = 0   � � ∂ 2 �� ∂t ( A eq u eq ) + ∂ + ∂ � � I d − µ 2 L [ A eq , d, σ ] ∂xI 1 ( x, A eq ) A eq u eq   ∂x  + µ 2 A eq Q [ A eq , d, σ ]( u eq ) = I 2 ( x, A eq ) + O ( µ 2 2 ) Remark A consistent one-parameter family (up to order O ( µ 2 2 ) ) can be introduced to improve the frequency dispersion. ◮ Bonneton, Barth´ elemy, Chazel, Cienfuegos, Lannes, Marche, and Tissier. European Journal of Mechanics-B/Fluids, 2011 ◮ Debyaoui, Ersoy. Part 2, preprint, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 15 / 21

A more stable formulation → useful for numerical purpose Define the linear T and the quadratic Q operators Define the averaged linear T and the quadratic Q operators Define the operators L and Q and finally the operator L Reformulated model  ∂tA eq + ∂ ∂   ∂x ( A eq u eq ) = 0   � � ∂ � ∂  + κ − 1 �� ∂t ( A eq u eq ) + ∂ 2 � � � I d − µ 2 κ L [ A eq , d, σ ] ∂xI 1 − I 2 A eq u eq ∂x κ  � ∂   + 1 �   + µ 2 A eq Q [ A eq , d, σ ]( u eq ) = O ( µ 2 ∂xI 1 − I 2 2 ) κ Remark A consistent one-parameter κ > 0 family (up to order O ( µ 2 2 ) ) can be introduced to improve the frequency dispersion. ◮ Bonneton, Barth´ elemy, Chazel, Cienfuegos, Lannes, Marche, and Tissier. European Journal of Mechanics-B/Fluids, 2011 ◮ Debyaoui, Ersoy. Part 2, preprint, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 15 / 21

Outline Outline 1 Derivation (based on Euler equations) 3D-2D 2D-1D 3D-1D 2 Improved model and stability Reformulated and stable models Invertible operator 3 Numerical analysis and test case Finite Volume scheme Numerical simulation 4 Conclusion and perspectives M. Ersoy (IMATH) 3D-1D 2020, 20 October 15 / 21

Invertibility of the operator T = A ( I d − µ 2 L [ A eq , d, σ ]) Theorem Let α , β and d ∈ C ∞ and A ∈ W 1 , ∞ ( R ) such that inf x ∈ R A ≥ A 0 > 0 . Then the b operator T : H 2 ( R ) → L 2 ( R ) is well-defined, one-to-one and onto. ◮ Debyaoui, Ersoy. Part 2, preprint, 2019 M. Ersoy (IMATH) 3D-1D 2020, 20 October 16 / 21

Invertibility of the operator T = A ( I d − µ 2 L [ A eq , d, σ ]) Theorem Let α , β and d ∈ C ∞ and A ∈ W 1 , ∞ ( R ) such that inf x ∈ R A ≥ A 0 > 0 . Then the b operator T : H 2 ( R ) → L 2 ( R ) is well-defined, one-to-one and onto. Let µ 2 ∈ (0 , 1) . Define the space H 1 µ 2 ( R ) the space H 1 ( R ) endowed with the norm � u � 2 µ 2 = � u � 2 2 + µ 2 � u x � 2 2 M. Ersoy (IMATH) 3D-1D 2020, 20 October 16 / 21

Invertibility of the operator T = A ( I d − µ 2 L [ A eq , d, σ ]) Theorem Let α , β and d ∈ C ∞ and A ∈ W 1 , ∞ ( R ) such that inf x ∈ R A ≥ A 0 > 0 . Then the b operator T : H 2 ( R ) → L 2 ( R ) is well-defined, one-to-one and onto. Let µ 2 ∈ (0 , 1) . Define the space H 1 µ 2 ( R ) Define the bilinear form a ( u, v ) a ( u, v ) = ( A T u, v ) = ( Au, v )+ � � � � �� √ √ A 3 A 3 µ 2 A √ − 2 d x u , √ − 2 d x v + ( Ad x u, d x v ) 3 u x 3 v x M. Ersoy (IMATH) 3D-1D 2020, 20 October 16 / 21

Invertibility of the operator T = A ( I d − µ 2 L [ A eq , d, σ ]) Theorem Let α , β and d ∈ C ∞ and A ∈ W 1 , ∞ ( R ) such that inf x ∈ R A ≥ A 0 > 0 . Then the b operator T : H 2 ( R ) → L 2 ( R ) is well-defined, one-to-one and onto. Let µ 2 ∈ (0 , 1) . Define the space H 1 µ 2 ( R ) Define the bilinear form a ( u, v ) Lax-Milgram theorem ∃ ! u ∈ H 1 µ 2 ( R ) ; a ( u, v ) = ( f, v ) , ∀ v ∈ H 1 µ 2 ( R ) , f ∈ L 2 ( R ) ⇓ ∃ ! u ∈ H 1 µ 2 ( R ) ; T u = f From definition of T , we get u xx = g ( A, u, d, σ ) ∈ L 2 ( R ) ⇒ u ∈ H 2 ( R ) . M. Ersoy (IMATH) 3D-1D 2020, 20 October 16 / 21

Outline Outline 1 Derivation (based on Euler equations) 3D-2D 2D-1D 3D-1D 2 Improved model and stability Reformulated and stable models Invertible operator 3 Numerical analysis and test case Finite Volume scheme Numerical simulation 4 Conclusion and perspectives M. Ersoy (IMATH) 3D-1D 2020, 20 October 16 / 21

Outline Outline 1 Derivation (based on Euler equations) 3D-2D 2D-1D 3D-1D 2 Improved model and stability Reformulated and stable models Invertible operator 3 Numerical analysis and test case Finite Volume scheme Numerical simulation 4 Conclusion and perspectives M. Ersoy (IMATH) 3D-1D 2020, 20 October 16 / 21

Numerical scheme : hyperbolic part We consider a classical Finite Volume scheme, U = ( A, Q ) i − δt n � � U n +1 = U n F i +1 / 2 ( U n i , U n i +1 ) − F i − 1 / 2 ( U n i − 1 , U n i ) i δx � 1 where F i ± 1 / 2 ≈ F ( U ( t, x i +1 / 2 )) dx is a Finite volume solver, δt n m i   Au � � with �   Au 2 + κ − 1 F ( U ) = I 1 − ′′ ′′ I 2 κ M. Ersoy (IMATH) 3D-1D 2020, 20 October 17 / 21

Numerical scheme : hyperbolic part We consider a classical Finite Volume scheme, U = ( A, Q ) i − δt n � � U n +1 = U n F i +1 / 2 ( U n i , U n i +1 ) − F i − 1 / 2 ( U n i − 1 , U n i ) i δx � 1 where F i ± 1 / 2 ≈ F ( U ( t, x i +1 / 2 )) dx is a Finite volume solver, for δt n m i instance, with upwind technique to deal with source term − s n F i ± 1 / 2 = F ( U ) + F ( V ) i 2 ( V − U ) 2   Au � � with �   Au 2 + κ − 1 F ( U ) = I 1 − ′′ ′′ I 2 κ ◮ Bourdarias, Ersoy, Gerbi. Journal of Scientific Computing, 2011 M. Ersoy (IMATH) 3D-1D 2020, 20 October 17 / 21

Numerical scheme : dispersive part We consider a classical Finite Volume scheme, U = ( A, Q ) i − δt n � � U n +1 = U n F i +1 / 2 ( U n i , U n i +1 ) − F i − 1 / 2 ( U n i − 1 , U n i ) i δx − δt n δx ([( I d − µ 2 L ) n ] − 1 D n ) i with ( D n ) i = D i +1 / 2 ( U n i − 1 , U n i , U n i +1 ) − D i − 1 / 2 ( U n i − 2 , U n i − 1 , U n i ) where D i ± 1 / 2 and [( I d − µ 2 L ) n ] − 1 are the centred approximation of � ∂ � D = 1 + µ 2 A Q and [( I d − µ 2 L )] − 1 ∂xI 1 − I 2 κ M. Ersoy (IMATH) 3D-1D 2020, 20 October 17 / 21

Numerical scheme : We consider a classical Finite Volume scheme, U = ( A, Q ) i − δt n � � U n +1 = U n F i +1 / 2 ( U n i , U n i +1 ) − F i − 1 / 2 ( U n i − 1 , U n i ) i δx − δt n δx ([( I d − µ 2 L ) n ] − 1 D n ) i Theorem The numerical scheme is stable under the classical CFL condition, λ ∈ Sp( D U F ( U )) | λ | δt n max δx � 1 . ◮ Debyaoui, Ersoy. NumHyp, 2020 M. Ersoy (IMATH) 3D-1D 2020, 20 October 17 / 21

Outline Outline 1 Derivation (based on Euler equations) 3D-2D 2D-1D 3D-1D 2 Improved model and stability Reformulated and stable models Invertible operator 3 Numerical analysis and test case Finite Volume scheme Numerical simulation 4 Conclusion and perspectives M. Ersoy (IMATH) 3D-1D 2020, 20 October 17 / 21

Propagation of a solitary wave ( κ = 1 ) Accuracy ( σ = d = 1 ) 2.2 2.19 2.18 2.17 M n 2.16 N = 200 2.15 N = 400 N = 800 2.14 N = 1600 N = 3200 2.13 0 2 4 6 8 10 12 14 16 18 20 t (s) Figure – M n := 0 ≤ i ≤ N +2 ( h n max i ) and M soliton ( t ) := 2 . 2 M. Ersoy (IMATH) 3D-1D 2020, 20 October 18 / 21

Propagation of a solitary wave ( κ = 1 ) Influence of the Section Variation ( N = 5000 cells) : σ ( x ; ε ) = β ( x ; ε ) − α ( x ; ε ) with � − ε 2 � x − L/ 2) 2 �� β = 1 2 − ε 2 exp and α = − β 2.215 2.21 2.205 2.2 M n 2.195 2.19 ε = 0 2.185 ε = 0.1 ε = 0.2 2.18 ε = 0.3 ε = 0.4 2.175 0 1 2 3 4 5 6 7 8 t (s) Figure – M n := x ∈ [0 ,L c ] ( h n max i ) M. Ersoy (IMATH) 3D-1D 2020, 20 October 18 / 21

Propagation of a solitary wave ( κ = 1 ) Influence of the Section Variation ( N = 5000 cells) : σ ( x ; ε ) = β ( x ; ε ) − α ( x ; ε ) with � − ε 2 � x − L/ 2) 2 �� β = 1 2 − ε 2 exp and α = − β 1.2 1.15 1.1 1.05 m n /m 0 1 0.95 ε = 0 0.9 ε = 0.1 ε = 0.2 0.85 ε = 0.3 ε = 0.4 0.8 0 1 2 3 4 5 6 7 8 t (s) N +1 Figure – Influence of σ : m n 1 m 0 with m n = � A n i N + 2 i =0 M. Ersoy (IMATH) 3D-1D 2020, 20 October 18 / 21

Propagation of a solitary wave ( κ = 1 ) Numerical order for ε = 0 N � η num − η exact � 2 � η num − η exact � ∞ 100 0.0789 0.0449 200 0.0497 0.0288 400 0.0304 0.0180 800 0.0198 0.0116 1600 0.0153 0.0081 3200 0.0138 0.0062 Order 0.53 0.58 M. Ersoy (IMATH) 3D-1D 2020, 20 October 18 / 21

Propagation of a solitary wave ( κ = 1 ) Numerical order for ε = 0 . 4 (reference solution obtained with N = 10000 cells) N � η num − η ref � 2 � η num − η ref � ∞ 100 0.05212 0.02533 200 0.02096 0.01082 400 0.01079 0.00554 800 0.00748 0.00503 1600 0.00635 0.00412 3200 0.00505 0.00300 Order 0.64 0.56 M. Ersoy (IMATH) 3D-1D 2020, 20 October 18 / 21

two solitary waves test case Comparison with the NLSW and the exact solution Figure – σ = 1 , d = 1 , N = 1000 , CFL = 0 . 95 , T f = 10 and κ = 1 . 159 M. Ersoy (IMATH) 3D-1D 2020, 20 October 19 / 21

two solitary waves test case Comparison with the NLSW and the exact solution Influence of κ 1.3 Exact solution κ =1 κ =1.159 κ =2 1.25 κ =7 NLWS 1.2 1.15 h 1.1 1.05 1 0 10 20 30 40 50 x (b) Solutions at time T f = 10 M. Ersoy (IMATH) 3D-1D 2020, 20 October 19 / 21

two solitary waves test case Comparison with the NLSW and the exact solution Influence of κ 0.016 L 1 Error 0.014 0.012 Error 0.01 0.008 0.006 0.004 1 2 3 4 5 6 7 κ (d) � h ex − h κ � 1 M. Ersoy (IMATH) 3D-1D 2020, 20 October 19 / 21