Spacetime Computing. A Dynamical Systems Approach to Turbulence. Bruce M. Boghosian. Department of Mathematics, Tufts University

Transcription

1 Spacetime Computing A Dynamical Systems Approach to Turbulence Bruce M. Boghosian, Tufts University Presented at 3rd NA-HPC Roadmap Workshop, Royal Society, London January 26, 2009 Acknowledgements: Hui Tang, Aaron Brown (Tufts Math), Spencer Smith (Tufts Physics) Luis Fazendeiro, Peter Coveney (UCL CCS) National Science Foundation

2 Outline Algorithmic: Spacetime computing, Parareal algorithm Application: Fluid turbulence Physics: Why turbulent flow difficult to simulate Mathematics: Dynamical systems, unstable periodic orbits Ongoing work: Spacetime computing for unstable orbits Lessons learned: Challenges and opportunities for NA/HPC January 26,

3 Not a new idea: Spacetime Computing J. Nievergelt, Comm. ACM 7 (1964) W. L. Miranker & W. Liniger, Math. Comp. 21 (1967) Both papers named Parallel methods for the numerical integration of ODEs. Experienced resurgence in 2001 with the Parareal Algorithm: J.-L., Lions, Y. Maday, G. Turinici, C.R. Acad Sci. Paris Sér. I Math 332 (2001) G. Bal, Y. Maday, In L.F. Pavarino and A. Toselli, eds., Recent Developments in domain Decomposition Methods, Vol. 23 of Lecture Notes in Computational Science and Engineering, Springer (2002) January 26,

4 The Parareal Algorithm Coarse gridpoints and fine gridpoints in time Predictor-corrector arrangement whereby results of fine-grid steps steer coarse-grid steps, and vice versa Evolution of coarse gridpoints is purely sequential Evolution of fine gridpoints can be done in parallel Achieves domain decomposition in time May be the only way to compute evolution equations faster than real time With periodic boundary conditions in time, it finds periodic orbits. January 26,

5 Example: ODE periodic orbits Initial-value problem: u t = Lu on t [0, T] with u(0) = u(t) Subdomain size τ = T/N Solve u (n) t = Lu (n) for t [nτ, (n + 1)τ] for n = 0,...,N 1. Boundary conds. u (n) (nτ) = λ n and u (n) (nτ + τ) = λ n+1 Periodicity λ 0 = λ N Next use coarse discretization, with grid step τ, to obtain coarse evolution equations for the λ n, given the u (n) (t). Iterate previous two steps. January 26,

6 The equations of motion of a fluid Viscous, incompressible flow The Navier-Stokes equations (Navier, 1823; Stokes, 1845) t u + u u = p + ν 2 u + F and u = 0 ν is the viscosity of the fluid (constant) Partial differential equations for velocity, pressure Only one free parameter Reynolds number : Re = Lu ν This is why wind tunnels work. January 26,

7 Laminar flow (low Re) M. van Dyke, Album of Fluid Motion, Parabolic Press (1982) January 26,

8 Periodic flow (intermediate Re) M. van Dyke, Album of Fluid Motion, Parabolic Press (1982) January 26,

9 Periodic flow in nature National Aeronautics and Space Administration January 26,

10 Turbulent flow (high Re) Reference: N.T. Ouellette, J.P. Gollub, Curvature Fields, Topology, and the Dynamics of Spatiotemporal Chaos, Phys. Rev. Lett. 99 (2007) Periodic flow in square domain Periodic force F = A sin(2πmx) sin(2πny)e x + A cos(2πmx) cos(2πny)e y Flow closely follows F for low Re Turbulent for high Re Show simulation January 26,

11 Problem: Algorithmic Complexity Grid-based algorithm Direct Numerical Simulation (DNS) Evolve one cubic unit of fluid for one unit of time Resolve smallest eddy: N x (Re) 3/4 Size of spatial grid: Nx 3 (Re) 9/4 Number of time steps: N t (Re) 3/4 Algorithmic Complexity C NxN 3 t (Re) 3 January 26,

12 State of the Art Best current DNS yields Re 10, , 000 Flow past aircraft requires Re 10, 000, , 000, 000 Desired increase in Re: 10 3 Required increase in computational power: 10 9 Hence Large Eddy Simulation (LES) but not a priori... No fully satisfactory a priori theory of turbulence exists today January 26,

13 Lorenz equations: Lorenz Attractor 8 >< >: ẋ = σ(y x) ẏ = xz + Rx y ż = xy bz State space is R 3 40 x, y, z t Much smaller-dimension state space than fluid, but has attractor, periodic orbits Attractor for Navier-Stokes equations in finite-dimensional (Constantin, Foias, Temam, 1985). January 26,

14 Unstable Periodic Orbits of Lorenz Attractor Correspond to any binary sequence (e.g., 11110) Dense in the attractor System is hyperbolic y 0 S 50. e T e F e z x 20 If you know all UPOs with period < T, you can make statistical predictions of any observable (DZF formalism). UPOs and their properties can be tabulated and stored in a curated database. January 26,

15 Results for 2d Navier-Stokes equations From LUPO code using Parareal algorithm Higher-order differencing needed Local spline fitting to orbit needed Method formulated using variational principle Show simulation January 26,

16 Symbol sequences and UPOs UPOs correspond to binary sequences Set of prime strings inequivalent to within cyclic shifts is the set of Lyndon words Length 1: 0, 1 Length 2: 01 Length 3: 001, 011 Length 4: 0001, 0011, 0111 Length 5: 00001, 00011, 00101, 00111, 01011, Lyndon words label Lorenz attractor UPOs January 26,

17 Symbolic dynamics of a turbulent fluid Boundaries determined by scalar functionals of velocity field e.g., energy, helicity We have extracted symbolic sequences Not a Markov partition, but still potentially useful Can we determine forbidden words from this sequence? Use learning algorithms e.g., hidden Markov models January 26,

18 Work in Progress I Time sequence Plot of t,t Clustering algs. HMM Buhl Kennel Initial guess for UPOs Conjugate gradient Orr Somerfeld equation Symbolic dynamics Kennel Mees UPOs Statistical error N 1 2 Pseudospectra Improved accuracy Monodromy matrix Dimension of attractor Ζ function Turbulent averages January 26,

19 Work in Progress II LUPO: Laboratory for unstable periodic orbits Shell models of turbulence (Tang, Boghosian) 2D Navier-Stokes (Lätt, Smith, Boghosian) 3D Navier-Stokes (Faizendeiro, Coveney, Boghosian) January 26,

20 Conclusions We are on the verge of an era in which the partial differential equations of the mathematical sciences may be simulated in spacetime Thanks to this, large-scale computation of hydrodynamic UPOs is becoming possible now. Improved understanding of fluid UPOs will soon lead to the first a priori statistical descriptions of turbulence This improved approach to turbulence modeling arose by taking results from numerical analysis, the mathematics of dynamical systems, and physics, and uniting people with expertise in each of these areas with computer scientists and software engineers. January 26,