AVARIATIONAL APPROACH FOR WATER WAVE MODELLING DENYS DUTYKH1 Senior Research Fellow UCD & Charge´ de Recherche CNRS 1University College Dublin School of Mathematical Sciences NUMERIWAVES Group Meeting ACKNOWLEDGEMENTS COLLABORATOR: Didier Clamond: Professor, LJAD, Universite´ de Nice Sophia Antipolis COLLABORATORS: ◮ Dimitrios Mitsotakis: University of California, Merced, USA ◮ Paul Milewski: University of Bath, UK WATER WAVE PROBLEM J. STOKER: WATER WAVES: THE MATHEMATICAL THEORY WITH APPLICATIONS [STO57] ◮ Continuity equation ∇2 x x,y φ = 0, ( , y) ∈ Ω × [−d,η], ◮ Kinematic bottom condition ∂φ + ∇φ ·∇d = 0, y = −d, ∂y ◮ Kinematic free surface condition ∂η ∂φ +∇φ·∇η = , y = η(x, t), ∂t ∂y ◮ Dynamic free surface condition ∇ ∂φ 1 2 η + |∇x,y φ| +gη+σ∇· = 0, y = η(x, t). ∂t 2 1 + |∇η|2 p HAMILTONIAN STRUCTURE A. PETROV (1964) [PET64]; V. ZAKHAROV (1968) [ZAK68] CANONICAL VARIABLES: η(x, t): free surface elevation φ˜(x, t): velocity potential at the free surface φ˜(x, t) := φ(x, y = η(x, t), t) EVOLUTION EQUATIONS: ∂η δH ∂φ˜ δH ρ = , ρ = − , ∂t δφ˜ ∂t δη HAMILTONIAN: η 1 2 1 2 2 H = |∇x,y φ| dy + gη + σ 1 + |∇η| − 1 Z 2 2 q −d LUKE’S VARIATIONAL PRINCIPLES J.C. LUKE, JFM (1967) [LUK67] First improvement of the classical Lagrangian (L := K − Π): t2 η 1 2 L = ρL dx dt, L := φt + |∇x,y φ| + gy dy Z Z Z 2 t1 Ω −d δφ: ∆φ = 0, (x, y) ∈ Ω × [−d,η], ∂φ ∇ ·∇ δφ|y=−d : ∂y + φ d = 0, y = −d, ∂η ∇ ·∇ ∂φ x δφ|y=η : ∂t + φ η − ∂y = 0, y = η( , t), ∂φ 1 ∇ 2 x δη: ∂t + 2 | φ| + gη = 0, y = η( , t). ◮ We obtain the water wave problem by varying η and φ GENERALIZATION OF THE LAGRANGIAN DENSITY D. CLAMOND &D.DUTYKH,PHYSICA D, (2012), [CD12] φ˜ := φ(x, y = η(x, t), t): quantity at the free surface φˇ := φ(x, y = −d(x, t), t): value at the bottom EQUIVALENT FORM OF LUKE’S LAGRANGIAN: η L ˜ ˇ 1 2 1 2 1 ∇ 2 1 2 = φηt + φdt − gη + gd − | φ| + φy dy 2 2 Z h2 2 i −d INTRODUCE THE VELOCITY FIELD: u = ∇φ, v = φy η 1 2 1 2 2 L = φη˜ t +φˇdt − gη − (u +v )+µ·(∇φ−u)+ν(φy −v) dy 2 Z−d h2 i GENERALIZATION OF THE LAGRANGIAN DENSITY D. CLAMOND &D.DUTYKH,PHYSICA D, (2012), [CD12] RELAXED VARIATIONAL PRINCIPLE: 1 L =(η +µ ˜ ·∇η − ν˜)φ˜ +(d +µ ˇ ·∇d +ν ˇ)φˇ − gη2 t t 2 η 1 2 1 2 + µ · u − u + νv − v +(∇· µ + νy )φ dy Z h 2 2 i −d CLASSICAL FORMULATION: η L ˜ ˇ 1 2 1 ∇ 2 1 2 = φηt + φdt − gη − | φ| + φy dy 2 Z h2 2 i −d DEGREES OF FREEDOM: η,φ; u, v ; µ,ν SHALLOW WATER REGIME CHOICE OF A SIMPLE ANSATZ IN SHALLOW WATER ◮ Ansatz: − u(x, y, t) ≈ u¯(x, t), v(x, y, t) ≈ (y + d)(η + d) 1 v˜(x, t) − φ(x, y, t) ≈ φ¯(x, t),ν(x, y, t) ≈ (y + d)(η + d) 1 ν˜(x, t) ◮ Lagrangian density: 1 2 1 2 1 1 2 L = φη¯ t − gη +(η + d) µ¯ · u¯ − u¯ + ν˜v˜ − v˜ − µ¯ · ∇φ¯ 2 h 2 3 6 i ◮ Nonlinear Shallow Water Equations: ht + ∇· [hu¯] = 0, u¯t +(u¯ ·∇)u¯ + g∇h = 0. CONSTRAINING WITH FREE SURFACE IMPERMEABILITY CONSTRAINT: ν˜ = ηt +µ ¯ ·∇η ◮ Generalized Serre equations [Ser53, GN76]: ht + ∇· [hu¯] = 0, 1 − u¯ + u¯ ·∇u¯ + g∇h + h 1∇[h 2γ˜] = (u¯ ·∇h)∇(h∇· u¯) t 3 −[u¯ ·∇(h∇· u¯)]∇h 2 γ˜ = v˜t + u¯ ·∇v˜ = h (∇· u¯) − ∇· u¯t − u¯ ·∇(∇· u¯) IT CANNOT BE OBTAINED FROM LUKE’S LAGRANGIAN: ∇ ¯ 1 ∇ ∇ ¯ δµ¯: u¯ = φ − 3 v˜ η 6= φ CONSTRAINING WITH FREE SURFACE IMPERMEABILITY CONSTRAINT: ν˜ = ηt +µ ¯ ·∇η ◮ Generalized Serre equations [Ser53, GN76]: ht + ∇· [hu¯] = 0, 1 − u¯ + u¯ ·∇u¯ + g∇h + h 1∇[h 2γ˜] = (u¯ ·∇h)∇(h∇· u¯) t 3 −[u¯ ·∇(h∇· u¯)]∇h 2 γ˜ = v˜t + u¯ ·∇v˜ = h (∇· u¯) − ∇· u¯t − u¯ ·∇(∇· u¯) SOLITARY WAVE SOLUTION (+ CNOIDAL WAVE): κ − η = a sech2 (x − ct), c 2 = g(d + a), (κd)2 = 3a(d + a) 1 2 SERRE EQUATIONS -I TWO-DIMENSIONAL CASE (1DH) ◮ Noncanonical Hamiltonian structure [Li02]: δH ht δq˜ ∂x h 0 1 3 = J , J = − , q˜ = hu¯− 3 (h u¯x )x q˜t δH ∂x q˜ + q˜∂x h∂x δh H = 1 hu¯2 + 1 h3u¯2 + gη2 dx 2 Z 3 x R ◮ Space translations, time translation and Galilean boost lead invariants: ηq˜ Q = dx, H , tq˜ − xη dx Z d + η Z R R SERRE EQUATIONS - II NUMERICAL DISCRETIZATION OF THE SERRE EQUATIONS ◮ The fully discrete 2nd order in space FV scheme: dh¯ 1 (1) (1) = − F (v¯) − F− (v¯) , dt ∆x + dv¯ ∆t (2) (2) (I − M) · = − F (v¯) − F− (v¯) + D(v¯). dt ∆x + ◮ Pseudo-spectral solver: ikd ikdF N(η, u¯) ηˆ + qˆ = −ikF{ηu¯}− , t 1 2 1 2 1 + 3 (kd) 1 + 3 (kd) 1 2 1 2 2 qˆt + ikg ηˆ = ikF 2 u¯ + 2 (d + η) u¯x − qu . REFERENCE [DCMM12]: D. Dutykh, D. Clamond, P. Milewski, D. Mitsotakis. Finite volume and pseudo-spectral schemes for the fully nonlinear ’irrotational’ Serre equations, 2012 SERRE EQUATIONS - III VALIDATION OF THE NUMERICAL SCHEME ◮ Exact solutions: 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 (x, 0) (x, 0) η 0.01 η 0.01 0 0 -0.01 -0.01 -0.02 -0.02 -0.03 -0.03 -40 -30 -20 -10 0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 x x (a) Solitary wave (b) Cnoidal wave SERRE EQUATIONS - III VALIDATION OF THE NUMERICAL SCHEME ◮ Convergence to the SW solution under the mesh refinement: 0 10 -1 10 -2 10 -3 10 ∞ ε -4 10 -5 10 -6 10 FV UNO2 N-2 Spectral method -7 10 1 2 10 10 N SERRE EQUATIONS - III VALIDATION OF THE NUMERICAL SCHEME ◮ Accuracy of invariants preservation: -2 -2 10 10 |H-H | |Q-Q | ex ex N-2 N-2 -3 -3 10 10 -4 -4 10 10 H(T) Q(T) -5 -5 10 10 -6 -6 10 10 1 2 3 1 2 3 10 10 10 10 10 10 N N (c) Hamiltonian H (d) Momentum Q SERRE EQUATIONS - III VALIDATION OF THE NUMERICAL SCHEME 0.4 0.2 (x,t) η 0 -0.2 40 20 0 10 0 20 -20 30 -40 40 t x HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06] −3 −3 x 10 x 10 Simulation Simulation 18 Experimental data 18 Experimental data 16 16 14 14 12 12 ) ) t 10 t 10 , , x x ( 8 ( 8 η η 6 6 4 4 2 2 0 0 −2 −2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (e) t = 18.5 s (f) t = 18.6 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06] −3 −3 x 10 x 10 Simulation Simulation 18 Experimental data 18 Experimental data 16 16 14 14 12 12 ) ) t 10 t 10 , , x x ( 8 ( 8 η η 6 6 4 4 2 2 0 0 −2 −2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (g) t = 18.7 s (h) t = 18.8 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06] −3 x 10 Simulation 18 Experimental data Simulation 0.025 Experimental data 16 14 0.02 12 ) ) t 10 0.015 t , , x x ( 8 ( η η 6 0.01 4 0.005 2 0 0 −2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (i) t = 18.92 s (j) t = 19.0 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06] Simulation Simulation 0.025 Experimental data 0.025 Experimental data 0.02 0.02 ) 0.015 ) 0.015 t t , , x x ( ( η 0.01 η 0.01 0.005 0.005 0 0 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (k) t = 19.05 s (l) t = 19.1 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06] −3 −3 x 10 x 10 Simulation Simulation 18 Experimental data 18 Experimental data 16 16 14 14 12 12 ) ) t 10 t 10 , , x x ( 8 ( 8 η η 6 6 4 4 2 2 0 0 −2 −2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (m) t = 19.15 s (n) t = 19.19 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al.