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M A S D O C Research Study Group: Data Assimilation for the 2D Navier–Stokes Equations Charles Brett, Andrew Lam, David McCormick, Michael Scott Mathematics and Statistics Centre for Doctoral Training (MASDOC), University of Warwick I I I I I Data assimilation is a technique used to make predictions of the future state of a system using information from both a model of the system and observations. It has many applications in fields such as geoscience, weather forecasting and hydrology. We focus on the simple case of applying 3DVar to 2D fluid flow with periodic boundary conditions, as there are still open questions here. Let {un} represent the state of the system at timestep n. Suppose we have model which can predict the state of the system at each timestep, and a sequence of noisy observations {yn}. In an ideal world, the model and the observations agree; how do we reconcile the model and the data when they do not agree? Goal: Estimate un given {y1, · · · , yn}, and only uncertain knowledge of u0. We consider the case when C −1 and Γ−1 are both powers of A, i.e., C −1 = Let X and Y be Hilbert spaces. If L is a self-adjoint, positive-definite operator on either X or Y , we define the L-inner product and L-norm as h·, ·iL := h·, L−1·i, k · kL := kL−1/2 · k. I I Let Ψ : X → X be the operator which evolves a system at time nh to the system at time (n + 1)h, with initial condition u0 ∈ X. We define {un} ⊂ X by un+1 := Ψ(un). Let {ξ¯n+1} denote a sequence of i.i.d. Y -valued random variables representing the observation error. We define the observations {yn} ⊂ Y by yn+1 := un+1 + ξ¯n+1. 1 β A . ε2 Γ−1 = I ûn+1 = η 2(I + η 2Aα )−1Aα Ψ(ûn) + (I + η 2Aα )−1yn+1, I (3) where α = γ − β and η = ε/δ. I I Here we present numerical simulations for 3DVar in the turbulent regime, with ν = 0.016, in the case α = −1. For frequent observations (i.e. for small h), we observe that two estimators with different initial conditions become asymptotically equal to each other and to the truth, even for large η (figure 2(a)). However, if we increase h from 0.032 to 0.16, we fail to get convergence unless η is made smaller (figure 2(b)). The spatial L2 difference between two estimators with different initial conditions converges exponentially to zero (figure 3(a)), which confirms theorem 1. After a time, the spatial L2 error between an estimator and the truth remains below a small threshold (figure 3(b)), which confirms theorem 2. Effects of α I I convergence to truth for k=(2,2) with η=150, h=0.16 and α=−1 convergence to truth for k=(2,2) with η=150, h=0.032 and α=−1 0.2 Theorem 1 (Convergence of estimators with same observations) v0 0.1 0 0 −0.1 −0.1 −0.2 0 2 4 6 8 10 12 −0.2 0 14 5 10 15 20 time time (a) The two estimators are asymptotically equal: for sufficiently large times they are almost indistinguishable. (b) Infrequent observations prevent the estimators locking onto the truth. Figure 2: Estimators exposed to the same observations but with two different initial conditions u0 and v0 . ⊗ denotes the estimators ûn and ◦ denotes Ψ(uˆn ), the estimators evolved forward by the model one time step. + denote the observations yn+1 that are being assimilated, and the black line is the truth. Theorem 2 (Convergence of estimators to the truth) L2 error between estimators for different ICs with η=25, h=0.032 and α=−1 0 −3 L2 error between estimators and truth with η=25, h=0.032 and α=−1 10 10 Suppose the truth un and the observation noise ξ¯n are uniformly bounded in H s; set M := sup kξ¯nks. Given initial estimators û0, v̂0 ∈ H s, for all sufficiently small η, there exists a constant λ < 1 such that, for all n ∈ N, −10 10 −4 −20 M kun − ûnks ≤ λ ku0 − û0ks + . 1−λ 10 10 n 3DVar 0.2 v0 0.1 We prove two theorems for the case α = −1, and perform numerical simulations for α = ±1. kûn − v̂nks ≤ λnkû0 − v̂0ks. u0 u0 If α = −1 the observations have a larger weight than the dynamics for large wavenumbers. If α = 1 the dynamics have a larger weight than the observations for large wavenumbers. Suppose the true solution un and the observation noise ξ¯n are uniformly bounded in H s := D(As/2) ⊂ H. Given initial estimators û0, v̂0 ∈ H s, for all sufficiently small η, there exists a constant λ < 1 such that, for all n ∈ N, 0.3 0.3 Equation (3) shows that ûn+1 is a convex combination of the dynamics Ψ(ûn) and the observations yn+1. α controls the weightings we put on each of these components: I I Notation 1 γ A , δ2 I Substituting this into equation (1) and multiplying through by ε2A−β we obtain I I Numerical results for α = −1 Analysis Introduction −30 I I I 10 Let ûn denote our estimate of un. Given ûn we define ûn+1 by 1 1 ûn+1 := arg min ku − yn+1k2Γ + ku − Ψ(ûn)k2C . 2 2 u∈X Numerical simulations −5 −40 This encodes our belief that the estimators should be guided by the observations (the 21 ku − yn+1k2Γ term) and the model (the 1 2 2 ku − Ψ(ûn )kC term). The estimator ûn+1 is the solution to 10 We used MATLAB, with code written by Kody Law, to simulate the NSE with different viscosity values ν. Here we exhibit typical stream functions from numerical simulations to show the limiting behaviour in three different regimes: steady, periodic, and turbulent dynamics. 0 2 4 6 8 10 12 10 14 0 2 4 6 8 10 12 14 time time (a) The squared L2 difference between estimators for two different initial conditions decreases exponentially; eventually we are limited by machine precision. (b) A plot of e2n := kûn − un k2 against time. The error decreases rapidly to around 10−4 . Figure 3: L2 error plots for α = −1. Stream function Stream function 0.05 0.3 0.8 0.8 0.04 (C −1 + Γ−1)ûn+1 = C −1Ψ(ûn) + Γ−1yn+1. 0.6 0.6 (1) 0.2 0.03 Numerical predictions for α = 1 0.4 0.4 0.02 0.2 0.01 0 x2 x2 0.2 The forward model 0.1 0 −0.2 −0.01 0 0 −0.2 −0.1 −0.4 We consider the 2D Navier-Stokes equations (NSE) on the torus T2: ∂tu − ν∆u + u · ∇u + ∇p = f ∇·u=0 u(x, 0) = u0(x) (2a) (2b) (2c) −0.03 −0.8 −0.04 −1 −1 −0.05 −0.8 −0.6 −0.4 −0.2 0 x1 0.2 I −0.8 0.4 0.6 0.8 −1 −1 −0.8 (a) ν = 0.04 (steady state) −0.6 −0.4 −0.2 0 x1 0.2 0.4 0.6 0.8 (b) ν = 0.03 (periodic state) convergence to truth for k=(2,2) with η=0.05, h=0.032 and α=1 0.8 0.8 0.6 0.6 du + νAu + B(u, u) = f , u(0) = u0, dt where A is the projection of −∆ onto H, B(u, u) is the projection of u · ∇u onto H, and, abusing notation, f is the forcing projected onto H. MASDOC CDT, University of Warwick 10 0.2 L2 error between estimators and truth with η=0.015, h=0.032 and α=1 0 10 u0 v0 0.4 0.2 L2 error between estimators for different ICs with η=0.015, h=0.032 and α=1 0 0.1 0.05 0.4 −10 10 −2 10 0 −20 10 0 0 −0.2 which projects the NSE into their divergence-free form, and use Fourier analysis. The NSE can be simplified to: I Here we present numerical simulations for 3DVar in the turbulent regime, with ν = 0.016, in the case α = 1. For well-chosen parameter values (e.g. η = 0.05, h = 0.032) two estimators with different initial conditions will converge together and to the truth (figure 4(a)). However η and h have to be chosen more carefully than in the α = −1 case to get convergence. The spatial L2 error between two estimators with different initial conditions will decrease exponentially to zero (figure 4(b)). After a time the spatial L2 error between an estimator and the truth will remain below a small threshold (figure 4(c)). This threshold appears to be smaller than in the α = −1 case. Stream function We work in the space Z H := u ∈ (L2(T2))2 : ∇ · u = 0, u(x) dx = 0 , T2 −0.2 −0.6 −0.3 2 I on T2 × (0, ∞) on T2 × (0, ∞) on T2 −0.6 I −0.4 x I −0.02 I −0.05 −4 −0.2 −0.4 −0.4 10 −30 10 −0.1 −0.6 −0.6 −0.8 −1 −1 −0.8 −0.8 −0.6 −0.4 −0.2 0 x1 0.2 0.4 0.6 −0.15 0 −6 −40 0.5 1 1.5 2 2.5 3 3.5 time 10 0 2 4 6 8 10 12 14 10 0 2 4 6 8 10 12 14 time time 0.8 (c) ν = 0.016 (turbulent state) Figure 1: Snapshots of the stream function of the NSE at one point in time, with different viscosity values. Mail: masdoc.info@warwick.ac.uk (a) The estimators become asymptotically equal, as in the α = −1 case. (b) As before, the square of the L2 difference between the estimators decreases exponentially. (c) We get a lower bound on the error e2n := kûn − un k2 than in the α = −1 case. Figure 4: Estimators and L2 error plots for α = 1. WWW: http://www2.warwick.ac.uk/fac/sci/masdoc/