A class of finite volume schemes for the 2D shallow water equations - PowerPoint PPT Presentation

A class of finite volume schemes for the 2D shallow water equations with Coriolis force E. Audusse 1 , V. Dubos 2 , A. Duran 3 , N. Gaveau 4 , Y. Nasseri 5 , Y.Penel 2 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inria–Paris, Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, France. 3 Institut Camille Jordan, Université Claude Bernard Lyon 1, France. 4 Institut Denis Poisson, Université d’Orléans, Université de Tours, CNRS, France. 5 Institut de Mathématique de Marseille, Aix-Marseille Université, France. August 22, 2019 CEMRACS 2019 V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 1 / 17

SW equations with Coriolis source term Ω being an open bounded domain of R 2 , flat bottom and T > 0 .  ∂ t h + ∇ · ( h u ) = in Ω × ( 0 , T ) , 0   − ω h u ⊥ , ∂ t ( h u )+ ∇ · ( h u ⊗ u )+ g h ∇ h = in Ω × ( 0 , T ) ,    u · n = on ∂ Ω × ( 0 , T ) , 0 h ( x , 0 ) = in Ω ,  h 0    u ( x , 0 ) = in Ω . u 0  Energy balance equation: ∂ t E + ∇ · (( 1 2 | u | 2 + gh ) h u ) = 0 , with E = 1 2 g h 2 + 1 2 h | u | 2 . V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 2 / 17

Linearised SW equations with Coriolis source term Linear equations around u 0 = 0 and h 0 >0: � ∂ t h + h 0 ∇ · u = in Ω × ( 0 , T ) , 0 − ω u ⊥ , ∂ t u + g ∇ h = in Ω × ( 0 , T ) . Energy balance equation: ∂ t E + ∇ · ( E u ) = 0 , with E = 1 2 g h 2 + 1 2 | u | 2 Geostrophic equilibrium: g ∇ h + ω u ⊥ = 0 . V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 3 / 17

Linearised SW equations with Coriolis source term Geostrophic equilibrium: g ∇ h + ω u ⊥ = 0 . Source : M. H. Do, Mathematical analysis of finite volume schemes for the simulation of quasi-geostrophic flows at low Froude number, 2017. V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 3 / 17

Outline Aim : 2D entropic scheme, consistent kernel with geostrophic equilibrium. State of art 1 Inaccuracy of the classic Godunov scheme Linearised SW with Coriolis source term Energy dissipative scheme for SW Collocated semi-discrete scheme 2 Modified equations Non-linear equations Linear equations Mixed semi-discrete scheme 3 Non-linear scheme Linear scheme V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 4 / 17

Inaccuracy of the classical Godunov scheme Modified equations:  ∂ t r + a ⋆ ∇ · u − κ r a ⋆ (∆ x ∂ 2 x r +∆ y ∂ 2 y r ) = 0  ∂ t u + a ⋆ ∇ x r − κ u a ⋆ ∆ x ∂ 2 = ω v x u ∂ t v + a ⋆ ∇ y r − κ u a ⋆ ∆ y ∂ 2 = − ω u y v  Source : E. Audusse, M. H. Do, P . Omnes, and Y. Penel [1] V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 5 / 17

Linearised SW with Coriolis source term [1] E. Audusse, M. H. Do, P . Omnes, and Y. Penel, Analysis of modified godunov type schemes for the two-dimensional linear wave equation with coriolis source term on cartesian meshes , JCP , 2018. Cell-centered semi-discrete scheme d r i , j ( t ) � � �� ∇ h r h + ω a ∗ u ⊥ + a ∗ [ ∇ h · u h ] i , j − ν r ∇ h · = 0 , h dt i , j d u i , j ( t ) = − ω u ⊥ + a ∗ [ ∇ h r h ] i , j − ν u [ ∇ h ( ∇ h · u h )] i , j i , j . dt preserves geostrophic equilibrium, ( ∇ h r h + ω a ∗ u ⊥ h = 0 ) = ⇒ ( ∇ h · u h = 0 ) , full discrete energy dissipation ( ν r = 0), vertex-based version. V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 6 / 17

Energy dissipative scheme for SW [2] F . Couderc, A. Duran and J.-P . Vila, An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification , JCP , 2017. Colocated explicit version K − ∆ t h n + 1 = h n F n m K ∑ e · � n e , K m e , K e ∈ ∂ K K − ∆ t � n e , K ) − � h n + 1 u n + 1 = h n K u n u n K ( F n n e , K ) + − u n K e ( F n m K ∑ e · � e · � m e K K e ∈ ∂ K � − ∆ t � h e = 1 h ∗ , n gh n K ∑ � 2 ( h K + h K e ) � n e , K m e . e � m K e ∈ ∂ K V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 7 / 17

Energy dissipative scheme for SW Colocated explicit version K − ∆ t h n + 1 = h n m K ∑ F n e · � n e , K m e , K e ∈ ∂ K K − ∆ t � n e , K ) − � h n + 1 u n + 1 = h n K u n m K ∑ u n K ( F n n e , K ) + − u n K e ( F n e · � e · � m e K K e ∈ ∂ K � h e = 1 � − ∆ t h ∗ , n gh n � K ∑ 2 ( h K + h K e ) � n e , K m e . e � m K e ∈ ∂ K F n e = ( h u ) n e − γ g Π n Π n � h n K e − h n � � , e − → ∇ e [ h ] = n e , K . e K h ∗ e = h n e − α Λ n Λ n � h u n K e − h u n � e − → ∇ e · [ h u ] = · � , n e , K . e K V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 7 / 17

Proposed scheme Semi-discrete scheme d h K � = − ∇ K · F dt d h K u K − ∇ up K · ( u , F ) − gh K ∇ K h − ω h K u ⊥ K + ω Π ⊥ = dt K where : | i | u K ( F K , i · e K , K i ) + + u i ( F K , i · e K , K i ) − � ∇ up ∑ � K · ( u , F ) = | K | i ∈{ e , w , n , s } ∇ K · F = F K , e − F K , w · e 1 + F K , n − F K , s · e 2 2 ∆ x 2 ∆ y ∇ K h = h e − h w e 1 + h n − h s 2 ∆ y e 2 2 ∆ x F K , i = 1 2 ( h K u K + h i u i − Π K − Π i ) Π K = γ ∆ t h k ( g ∇ K h + ω u ⊥ K ) V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 8 / 17

Modified equations associated scheme Semi-discrete scheme d � = − ∇ K · h u K + ∇ K · Π K dt h k − ∇ up d K · ( u , F ) − gh K ∇ K h − ω h u ⊥ K + ω Π ⊥ = dt h K u K K with Π K = γ ∆ t h k ( g ∇ K h + ω u ⊥ K ) Modified non linear equations � ∂ t h = − ∇ · h u + ∇ · Π − ∇ · ( F ⊗ u ) − g h ∇ h − ω h u ⊥ + ω Π ⊥ ∂ t ( h u ) = where : Π = γ ∆ t ( g h ∇ h + ω h u ⊥ ) . Mechanic energy balance of the modified equations : � � 1 ( gh + 1 2 ( gh 2 + h u 2 ) � 2 . � g h ∇ h + ω h u ⊥ � 2 | u | 2 ) h u � � � ∂ t + ∇ · = − γ ∆ t V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 9 / 17

Energy of the scheme � � d 1 � � = ∑ gh K ∇ K · Π + ω Π ⊥ 2 ( gh 2 K + h K | u | 2 ∑ K ) | K | K · u K + R K dt K ∈ T K ∈ T where R K = 1 1 | e |� u Ke − u K � 2 ( F e · n e , K ) − ≤ 0 | K | ∑ 2 e ∈ ∂ K K u K ) = − ∑ ( gh K ∇ K · Π + ω Π ⊥ (( g ∇ K h + ω u ⊥ ∑ K ) · Π K ) ≤ 0 K ∈ T K ∈ T Π K = γ ∆ t ( g h K ∇ K h + ω h u ⊥ ) . thanks to the grad-div duality, preserved by our discretisations Energy decreasing property � � d 1 2 ( gh 2 K + h K | u | 2 ∑ K ) ≤ 0 dt K ∈ T V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 10 / 17

Linearisation of the scheme Semi-discrete linearised scheme around h 0 > 0 and u 0 = 0 d dt h K + ∇ K · ( h 0 u K − Π K ) = 0, d − ω ( h 0 u K − Π K ) ⊥ . dt u K + g h 0 ∇ K h = h 0 where : Π K = h 0 γ ∆ t ( g ∇ K h + ω u ⊥ K ) . Remark : Very similar to Audusse and al [1]. � ∂ t h + ∇ · F = 0 , Modified equations associated: − ω F ⊥ , ∂ t h 0 u + g h 0 ∇ h = where: F = h 0 u − γ ∆ t h 0 ( g ∇ h + ω u ⊥ ) . Geostrophic equlibrium : g ∇ h + ω u ⊥ = 0 V. Dubos, N. Gaveau, Y. Nasseri ( 1 Université Paris 13, LAGA, CNRS, UMR 7539, Institut Galilée Villetaneuse, France. 2 Team ANGE, Inr FV schemes for Shallow Water with Coriolis force CEMRACS 2019 August 22, 2019 11 / 17