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

CIRM Luminy, September 28 – October 2, 2009

Studying the asymptotic structure of solutions

• f hydrodynamical equations
• W. Pauls

Max Planck Institute for Dynamics and Self-Organization, G¨

• ttingen, 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) ∂tu + u ∂xu = −να(−∂2

x)αu

– Well-posedness ∗ Three-dimensional Euler equation ∂tu + 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 ∂tu + u · ∇u = −∇p − µDu, ∇ · 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−αe−δr

• n a regular 1D grid r0, 2r0, ..., Nr0

Gn = G(nr0), 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)

Rn ≃ GnGn−2 G2

n−1

=

• 1 −

1 (n − 1)2 −α Ignore subleading corrections α = − ln Rn 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 Gn in the “most asymptotic” region n = L, ..., N
• Transformations:

I Inverse: Gn − →

1 Gn

R Ratio: Gn − →

Gn Gn−1

SR Second ratio: Gn − → GnGn−2

G2

n−1

D Difference: Gn − → Gn − Gn−1

• Going down (assuming Gn > 0):

– Test 1: if Gn < 1 apply I – Test 2: Does Gn grow faster than n5/2? ∗ 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

∂tu + u (∂xu) = 0, u(0, x) = u0(x) = −(1/2) sin x – Fourier–Lagrangian representation (Platzman, Fournier–Frisch) u(t, x) =

• k=±1, ±2,...

eikxˆ uk(t), ˆ uk(t) = − 1 2iπkt 2π e−ik(a+tu0(a))da – Explicit solution ˆ

uk(t) =

1 iktJk(kt/2) in terms of the Bessel function Jk of order k gives

via asymptotic (Debye) expansion

ˆ uk(t) ∼ 1 it 1

• 2π
• 1 − (t/2)2

k− 3

2 e−k(arccosh 2 t −√

1−(t/2)2)

 1 +

∞

• n=1

γn(

1

√

1−(t/2)2 )

kn   ,

where γn 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

1.00004 1.00008 1.00012 1.00016 200 400 600 800 1000

Gk

(1)

1e-06 2e-06 3e-06 4e-06 200 400 600 800 1000

Gk

(2)

1e+08 2e+08 3e+08 4e+08 200 400 600 800 1000

Gk

(3)

2e+05 4e+05 6e+05 8e+05 1e+06 200 400 600 800 1000

Gk

(4)

500 1000 1500 2000 200 400 600 800 1000

Gk

(5)

1e-10 1e-08 1e-06 1e-04 200 400 600 800 1000

Gk

(6) - 2

Ck−αe−δk

SR

− → 1 + α k2

−D

− → 2α k3

I

− → k3 2α

D

− → 3k2 2α

D

− → 3k α

D

− → 3 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 6 stages 1.49999999993 0.4509324931404 0.4286913791 13 stages 1.49999999999999995 0.450932493140378061868 0.4286913790524959

• Theor. value

3/2 0.450932493140378061861 0.42869137905249585643 γ1 γ2 γ3 6 stages −0.17641252 0.17295 −0.401 13 stages −0.17641258225238 0.172968106990 −0.406446182

• Theor. value

−0.176412582252385 0.1729681069958 −0.4064461802 γ4 γ5 γ6 13 stages 1.384160933 −6.192505762 34.5269751

• Theor. value

1.3841609326 −6.192505761856 34.5269752864

8

Nonuniversal nature of complex singularities of the Euler equation

• Euler equation in 2D

∂t∇2Ψ − J(Ψ, ∇2Ψ) = 0,

• Short-time asymptotic r´

egime: time-independent, one initial condition is enough Ψ0(z1, z2) = ˆ F(p)e−ip·z + ˆ F(q)e−iq·z

• Translation invariance =

⇒ ˆ F(p) = 1, ˆ 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 G(k) ≃ C(θ)k− 5

2 e−δ(θ)k

• 1 + b1(θ)

k + a2(θ) ln k k2 + O 1 k2

• 1e-08

1.5e-08 2e-08 2.5e-08 3e-08 3.5e-08 4e-08 4.5e-08 5e-08 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 θ Discrepancy

10

1D Burgers equation with exponentially growing dissipation

• Burgers equation in the Fourier space

∂tˆ u(k, t) + ik 2

• p∈Z

ˆ u(p, t)ˆ u(k − p, t) = −e|k|ˆ u(k, t).

• A dominant balance argument suggests for |k| → ∞

ˆ u(k, t) ∼ e−

1 ln 2 |k| ln |k|

• Numerical solution

– Consider complex-valued initial conditions such that ˆ u0(k) = 0 for k ≤ 0 and k > K – Observation: all Fourier coefficients can be calculated by a finite number of

• perations

ˆ u(k, t) = e−teku0(k) − ik 2 t ds e−(t−s)ek k−1

• p=1

ˆ u(p, s)ˆ u(k − p, s) – Number of operations grows with k as

1 4k √ 3eπ√ 2k/3 11

1D Burgers equation with exponentially growing dissipation cont’d

• Identifying the decay rate: asymptotic extrapolation

– Consider the array ˆ u(k, t) = g0(k) for fixed t – Transformations: e−Ck ln k

Log

− → −Ck ln k

D

− → −C ln k

D

− → −C 1 k

I

− → − k C

D

− → − 1 C

?

= − ln 2

• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02 0.025 8 10 12 14 16 18 20 22 24 Discr (k) k

12

A simplified model for the 1D Burgers equation

• For the simple initial condition u0(x) = iAe−ix the solution becomes

ˆ u(k, t) = iAkg(k, t)

• Coefficients are calculated recursively

– The first coefficients are g(1, t) = e−e1t, g(2, t) = e−2e1t e2 − 2e1 − e−e2t e2 − 2e1 – Generally g(k, t) =

• ω∈P (k)

hk(ω) exp

• −t
• i

eωi where ∗ P(k) are all integer partitions of k ∗ ω = {ω1, ..., ωi, ...} is an integer partition of k ∗ hk(ω) is a numerical coefficient

• For t → ∞ terms with ωi = 1 dominate, so that g(k, t) ≃ f(k) exp(−tke1)

13

A simplified model for the 1D Burgers equation cont’d

• The coefficients f(k) are obtained from the recursion relation

f(k) = k 2 1 ek − ke1

k−1

• l=1

f(l)f(k − l), f(1) = 1

• Asymptotic extrapolation does not work: use dominant balance and fitting instead

f(k) ≃ 1 2

3 2 √

π ln 2 k− 3

2 eδke− 1 ln 2 k ln k exp

• ˜

δk sin 2π ln 2 ln k + φ

• 1.3579
• 1.35785
• 1.3578
• 1.35775
• 1.3577
• 1.35765
• 1.3576
• 1.35755
• 1.3575

100 1000 10000 k Rate of exponential decay

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