Variational Assimilation of Discrete Navier-Stokes Equations

Variational Assimilation of Discrete Navier-Stokes Equations Souleymane.Kadri-Harouna FLUMINANCE, INRIA Rennes-Bretagne Atlantique Campus universitaire de Beaulieu, 35042 Rennes, France

Outline Discretization of Navier-Stokes Equations Temporal discretization Spatial discretization

Outline Discretization of Navier-Stokes Equations Temporal discretization Spatial discretization Variational Assimilation Principle Discrete adjoint Method

Outline Discretization of Navier-Stokes Equations Temporal discretization Spatial discretization Variational Assimilation Principle Discrete adjoint Method Preliminary resultats

Outline Discretization of Navier-Stokes Equations Temporal discretization Spatial discretization Variational Assimilation Principle Discrete adjoint Method Preliminary resultats Conclusion and Outlook

Incompressible Navier-Stokes Equations Cauchy problem for Navier-Stokes: t v ν v + (v )v + p = f, x Ω, t [0, T ], (NS) v = 0, x Ω, t [0, T ], v(0, x) = v 0 (x), x Ω. Unknowns : velocity v(t, x) and pressure p(t, x) Projecting the system (NS) onto H div (Ω) 1 yields: t v = ν v + P[ (v )v + f] (NSP) with P orthogonal projector from (L 2 (Ω)) d to H div (Ω). The pressure p is recovered through the Helmholtz decomposition: p = (v )v + f P[ (v )v + f] 1 (L 2 (Ω)) d divergence-free function space

Helmholtz-Decomposition The projector P is explicite in Fourier domain: u (L 2 (Ω)) d, û(ξ) = ξ (1 ξt ξ 2 û(ξ) + ξ ) ξt ξ 2 û(ξ) Thus P(u)(ξ) = (1 ξ ξt ξ 2 ) û(ξ) For space localization and adaptativity: Periodic Anisotropic divergence-free wavelets [Deriaz,Perrier 08] For physical boundary conditions: Anisotropic divergence-free wavelets [Kadri-Harouna,Perrier 10]

Temporal discretization Heat kernel integration problem: t v ν v = f, with f = P[ (v )v + f]. Implicite finite difference approximation v(x, nδt) v n : v n+1 v n δt ν 2 (vn+1 + v n ) = f n, Crank-Nicholson O(δt 2 ) Heat kernel factorization (ADI method): ) ) ) (1 α 2 x 2 α 2 y 2 = (1 α (1 2 x 2 α 2 y 2 + O(α 2 )

Spatial discretization Semi implicite treatment for the non-linear term: (v n+1/2 )v n+1/2 = 3 2 (vn )v n 1 2 (vn 1 )v n 1 CFL condition: δt C (δx/v max ) 4/3. Scale separation: v(t, x) = j,k d j,k div (t) Ψdiv j,k (x) ODE system on the coefficients [d div j,k (t)]. Galerkin method in space with V j = (V 1 j V 0 j ) (V 0 j V 1 j ). At each time step we need to compute the projector P.

Principle of Variational Assimilation Measurements (observations) denoted v k ob, k = 1,, N Discrete dynamical model equation: with L 1/2 v n+1 L 1/2 v n + 3 2 Bn v n 1 2 Bn 1 v n 1 = 0, L 1/2 := 1 δt 2, L 1/2 := 1 + δt 2, Bn v n := P(v n )v n Objective: find the most probable state defined both by the measurements and dynamical equations. Cost function minimization: J(v 0 ) = 1 2 N H v k v k ob 2 δt + α 2 v 0 2, k=1

Differentiation operators Let f : E R be a vector (or scalar) function Directional derivative: if the following limit exists d f (v) := lim h 0 f (v + hd) f (v) h Fréchet derivative: if there exist f (v) E such that f (v + u) = f (v) + f (v), u + o( u ) If the gradient of f exists, then: d f (v) = f (v), d

Cost function differentiation J(v 0 + hu) J(v 0 ) = 1 2 1 2 N H v k (v 0 + hu) v k ob 2 δt + α 2 v 0 + hu 2 k=1 N H v k v k ob 2 δt α 2 v 0 2 k=1 Rewritten the terms, we get: H v k (v 0 + hu) v k ob 2 H v k v k ob 2 = + H v k (v 0 + hu) + H v k 2v k ob, H v k (v 0 + hu) H v k and v 0 + hu 2 v 0 2 = 2v 0 + hu, hu Thus: N uj(v 0 ) = H v k v k ob, H v k uv k δt + α v 0, u k=1

Problem Let us consider the one-dimensional ODE t y = F (t).y, with F (t) a linear operator The continuous adjoint model is t λ = F (t) T.λ Discretizing with an explicit Euler scheme, we get: y n+1 y n = δtf n.y n y n+1 = (1 + δtf n ).y n For which we get Otherwise: y n = (1 + δtf n ) T.y n+1 λ n λ n+1 = δtf T n+1.λ n+1 λ n = (1 + δtf n+1 ) T.λ n+1 (1 + δtf n+1 ) T (1 + δtf n ) T

Linear tangent One disturbs the initial condition: ṽ 0 = v 0 + hu. Then, we get: L 1/2 ṽ n+1 L 1/2 ṽ n + 3 2 B n ṽ n 1 2 B n 1 ṽ n 1 = 0. Taking the difference with the non disturbed equation, we have: L 1/2 (ṽ n+1 v n+1 ) L 1/2 (ṽ n v n ) = 3 2 ( B n ṽ n B n v n ) + 1 2 ( B n 1 ṽ n 1 B n 1 v n 1 ) Multiplying with 1/h and taking the limit as h 0, we get: L 1/2 u v n+1 L 1/2 u v n = 3 2 Bn v n u v n + 1 2 Bn 1 v n 1 u v n 1

Adjoint variable Taking the inner product of the linear tangent with λ n+1 yields u v n+1, L 1/2 λ n+1 u v n, L 1/2 λ n+1 = u v n, 3 2 B n λ n+1 + u v n 1, 1 2 B n 1 λ n+1 Thus, making identification, the adjoint model is defined as: L 1/2 λ N = F N, L 1/2 λ N 1 L 1/2 λ N + 3 2 B N 1 λ N = F N 1 with F n = H v n (H v n v n ob), 1 n N For 1 n N 2, L 1/2 λ n L 1/2 λ n+1 + 3 2 B n λ n+1 1 2 B n λ n+2 = F n J(v 0 )= αv 0 + L 1/2 λ 1 3 2 B 0 λ 1 + 1 2 B 0 λ 2

Model error 6.8 x 10 8 6.7 6.6 L 2 Norm 6.5 6.4 6.3 6.2 1 2 3 4 5 6 7 8 9 10 Time Figure: L 2 -norm error

Error on real experience Two types of observation H v k v k ob := v k I v k ob, v k ob Optical-Flow H v k v k ob := I k 1 (x + v k ) I k 0 (x)

Pseudo observations error (a) True vorticity (b) Estimated vorticity Figure: Optical-flow observation: RMSE=0.0544.

Pseudo observations error Figure: RSE on the vorticity = 0.0103

DFD observations error (a) True vorticity (b) Estimated vorticity Figure: DFD observation: RMSE=0.0696, j = 7.

DFD observations error Figure: RSE on the vorticity

DFD observations and diffusion (a) True vorticity (b) Estimated vorticity Figure: DFD observation: RMSE=0.0523,ν 2.73E 5, Re 1 1.33E 6.

Conclusion and Outlook Navier-Stokes discretization Discrete adjoint models

Conclusion and Outlook Navier-Stokes discretization Discrete adjoint models Models with low complexity

Conclusion and Outlook Navier-Stokes discretization Discrete adjoint models Models with low complexity Wavelet adaptativity in the simulation Use methods on a dynamic geophysical flow models