Studying the asymptotic structure of solutions of hydrodynamical - PowerPoint PPT Presentation

CIRM Luminy, September 28 – October 2, 2009 Studying the asymptotic structure of solutions of hydrodynamical equations W. Pauls Max Planck Institute for Dynamics and Self-Organization, G¨ ottingen, Germany Models • One-dimensional inviscid Burgers equation • Two-dimensional incompressible Euler equation • One-dimensional Burgers equation with hyperviscosity/exponential dissipation W. Pauls and U. Frisch, J. Stat. Phys. 127 (2007) 1095–1119 W. Pauls, submitted to Physica D (2009) A. Gilbert and W. Pauls, submitted to Nonlinearity (2009) C. Bardos, U. Frisch, W. Pauls, S. S. Ray and E. S. Titi, Communications in Mathematical Physics (2009) in press 1

Motivations • Properties of hydrodynamical PDEs – Integrability of equations ∗ Burgers equation with normal ( α = 1 ) and hyperdissipation ( α > 1 ) ∂ t u + u ∂ x u = − ν α ( − ∂ 2 x ) α u – Well-posedness ∗ Three-dimensional Euler equation ∂ t u + u · ∇ u = −∇ p, ∇ · u = 0 • Analytic properties of solutions – Existence of real or complex singularities – Type of singularities ∗ Navier–Stokes equation with different kinds of dissipation ∂ t u + u · ∇ u = −∇ p − µ D u , ∇ · u = 0 2

How to determine asymptotics in practice • It is difficult to obtain asymptotic expansions by theory • Numerical approach – Solutions computed with high precision – Methods: integral transforms, series expansions – Numerical determination of the asymptotics 3

Asymptotic interpolation/extrapolation: 1D • Given a function G ( r ) with assumed leading-order expansion ( r → ∞ ) G ( r ) ≃ Cr − α e − δr on a regular 1D grid r 0 , 2 r 0 , ..., Nr 0 G n = G ( nr 0 ) , n = 1 , 2 , ..., N can we determine C , α and δ numerically with high accuracy? What about subleading terms? • Naive method: least square fit • Improvement: take second ratio (Shelley, Caflisch, Pauls–Matsumoto–Frisch–Bec) � − α � R n ≃ G n G n − 2 1 = 1 − G 2 ( n − 1) 2 n − 1 Ignore subleading corrections ln R n α = − ln(1 − 1 / ( n − 1) 2 ) • Is there a more systematic approach? 4

Joris van der Hoeven’s asymptotic extrapolation J. van der Hoeven, On asymptotic extrapolation, J. Symbolic Computation 44 (2009) • Interpolate the sequence G n in the “most asymptotic” region n = L, ..., N • Transformations: 1 I Inverse : G n − → G n G n R Ratio : G n − → G n − 1 → G n G n − 2 SR Second ratio : G n − G 2 n − 1 D Difference : G n − → G n − G n − 1 • Going down (assuming G n > 0 ): – Test 1: if G n < 1 apply I – Test 2: Does G n grow faster than n 5 / 2 ? ∗ Yes: if the growth is exponential apply SR , otherwise R ∗ No: apply D – Continue untill obtaining data which are easy to interpolate and clean enough • Going up: invert the transformations I , R , SR and D 5

Implementing asymptotic extrapolation for the Burgers equation with single-mode initial condition • Inviscid Burgers equation ∂ t u + u ( ∂ x u ) = 0 , u (0 , x ) = u 0 ( x ) = − (1 / 2) sin x – Fourier–Lagrangian representation (Platzman, Fournier–Frisch) � 2 π 1 � e ikx ˆ e − ik ( a + tu 0 ( a )) da u ( t, x ) = u k ( t ) , u k ( t ) = − ˆ 2 iπkt 0 k = ± 1 , ± 2 ,... 1 – Explicit solution ˆ u k ( t ) = ikt J k ( kt/ 2) in terms of the Bessel function J k of order k gives via asymptotic (Debye) expansion 1 √   γ n ( 1 − ( t/ 2) 2 ) t − √ ∞ u k ( t ) ∼ 1 1 k − 3 2 e − k (arccosh 2 � 1 − ( t/ 2) 2 )  , ˆ  1 + k n it � � 1 − ( t/ 2) 2 2 π n =1 where γ n are known polynomials (see e.g. Abramowitz–Stegun) • Testing asymptotic extrapolation: Fourier coefficients calculated numerically with high precision (80 digits) at t = 1 and high resolution | k | ≤ 1000 . 6

4e-06 1.00016 (2) G k (1) G k 3e-06 1.00012 2e-06 1.00008 1e-06 1.00004 0 200 400 600 800 1000 200 400 600 800 1000 4e+08 1e+06 (3) (4) G k G k 8e+05 3e+08 6e+05 2e+08 4e+05 1e+08 2e+05 0 0 200 400 600 800 1000 200 400 600 800 1000 2000 (5) G k (6) - 2 G k 1e-04 1500 1e-06 1000 1e-08 500 1e-10 200 400 600 800 1000 200 400 600 800 1000 → k 3 → 3 k 2 → 1 + α → 2 α → 3 k → 3 − D SR I D D D Ck − α e − δk − − − − − − k 2 k 3 2 α 2 α α α 7

Burgers equation: comparing numerical and theoretical results • The extrapolation procedure can be continued to improve accuracy on α , δ and to determine more subdominant terms – Transformations applied: SR, -D, I, D, D, D, D, I, D, D, D, D, D α δ C 1 . 49999999993 0 . 4509324931404 0 . 4286913791 6 stages 1 . 49999999999999995 0 . 450932493140378061868 0 . 4286913790524959 13 stages 3 / 2 0 . 450932493140378061861 0 . 42869137905249585643 Theor. value γ 1 γ 2 γ 3 − 0 . 17641252 0 . 17295 − 0 . 401 6 stages − 0 . 17641258225238 0 . 172968106990 − 0 . 406446182 13 stages − 0 . 176412582252385 0 . 1729681069958 − 0 . 4064461802 Theor. value γ 4 γ 5 γ 6 1 . 384160933 − 6 . 192505762 34 . 5269751 13 stages 1 . 3841609326 − 6 . 192505761856 34 . 5269752864 Theor. value 8

Nonuniversal nature of complex singularities of the Euler equation • Euler equation in 2D ∂ t ∇ 2 Ψ − J (Ψ , ∇ 2 Ψ) = 0 , • Short-time asymptotic r´ egime: time-independent , one initial condition is enough F ( p ) e − i p · z + ˆ Ψ 0 ( z 1 , z 2 ) = ˆ F ( q ) e − i q · z ⇒ ˆ F ( p ) = 1 , ˆ • Translation invariance = F ( q ) = 1 . • Parameters: ratio of moduli of the basic vectors η = | q | / | p | and the angle φ between p and q • Vorticity diverges near the singularities ( s is the distance to the singular set) ω ∼ s − β , • Exponent β depends on φ (but not on η ) and is thus nonuniversal ! • In particular, β ( φ ) → 1 when φ → 0 and β ( φ ) → 1 / 2 when φ → π . 9

Case φ → 0 : precise determination of the nature of singularities • Numerically determined asymptotic expansion ( k = | k | (cos θ, sin θ ) , | k | → ∞ ) of the stream function in the Fourier space � 1 � �� 1 + b 1 ( θ ) + a 2 ( θ ) ln k G ( k ) ≃ C ( θ ) k − 5 2 e − δ ( θ ) k + O k 2 k 2 k 5e-08 Discrepancy 4.5e-08 4e-08 3.5e-08 3e-08 2.5e-08 2e-08 1.5e-08 1e-08 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 θ 10

1D Burgers equation with exponentially growing dissipation • Burgers equation in the Fourier space u ( k, t ) + ik � u ( k − p, t ) = − e | k | ˆ ∂ t ˆ u ( p, t )ˆ ˆ u ( k, t ) . 2 p ∈ Z • A dominant balance argument suggests for | k | → ∞ 1 u ( k, t ) ∼ e − ln 2 | k | ln | k | ˆ • Numerical solution – Consider complex-valued initial conditions such that ˆ u 0 ( k ) = 0 for k ≤ 0 and k > K – Observation: all Fourier coefficients can be calculated by a finite number of operations � t ds e − ( t − s ) e k k − 1 u ( k, t ) = e − te k u 0 ( k ) − ik � ˆ u ( p, s )ˆ ˆ u ( k − p, s ) 2 0 p =1 3 e π √ 1 2 k/ 3 – Number of operations grows with k as √ 4 k 11

1D Burgers equation with exponentially growing dissipation cont’d • Identifying the decay rate: asymptotic extrapolation – Consider the array ˆ u ( k, t ) = g 0 ( k ) for fixed t – Transformations: → − C 1 → − k → − 1 Log D D I D ? e − Ck ln k − → − Ck ln k − → − C ln k − − − = − ln 2 k C C 0.025 0.02 0.015 0.01 Discr (k) 0.005 0 -0.005 -0.01 -0.015 8 10 12 14 16 18 20 22 24 k 12

A simplified model for the 1D Burgers equation • For the simple initial condition u 0 ( x ) = iAe − ix the solution becomes u ( k, t ) = iA k g ( k, t ) ˆ • Coefficients are calculated recursively – The first coefficients are e − 2 e 1 t e − e 2 t g (1 , t ) = e − e 1 t , g (2 , t ) = e 2 − 2 e 1 − e 2 − 2 e 1 – Generally � e ω i � � � g ( k, t ) = h k ( ω ) exp − t i ω ∈ P ( k ) where ∗ P ( k ) are all integer partitions of k ∗ ω = { ω 1 , ..., ω i , ... } is an integer partition of k ∗ h k ( ω ) is a numerical coefficient • For t → ∞ terms with ω i = 1 dominate, so that g ( k, t ) ≃ f ( k ) exp( − tke 1 ) 13

A simplified model for the 1D Burgers equation cont’d • The coefficients f ( k ) are obtained from the recursion relation k − 1 f ( k ) = k 1 � f ( l ) f ( k − l ) , f (1) = 1 e k − ke 1 2 l =1 • Asymptotic extrapolation does not work: use dominant balance and fitting instead � 2 π 1 ln 2 k ln k exp � �� k − 3 1 ˜ 2 e δk e − 2 √ f ( k ) ≃ δk sin ln 2 ln k + φ 3 2 π ln 2 -1.3575 Rate of exponential decay -1.35755 -1.3576 -1.35765 -1.3577 -1.35775 -1.3578 -1.35785 -1.3579 100 1000 10000 k 14

Conclusions • For solutions with “well-behaved” Fourier coefficients the asymptotic extrapolation method can be used successfully • Examples: – Riemann equation (inviscid Burgers equation) – Two-dimensional Euler equation for a special initial condition • If the solutions are not “well-behaved” this method does not work • Examples: – Hyperviscous Burgers equation – Burgers equation with exponentially growing dissipation 15