Symplectic Integrators for Klein-Gordon chains - Recurring formation - - PowerPoint PPT Presentation

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Symplectic Integrators for Klein-Gordon chains - Recurring formation - - PowerPoint PPT Presentation

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Symplectic Integrators for Klein-Gordon chains - Recurring formation of localized, breather-like oscillations Efstratios-Georgios Efstratiadis March 29, 2020 Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Outline

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Symplectic Integrators for Klein-Gordon chains - Recurring formation of localized, breather-like oscillations

Efstratios-Georgios Efstratiadis March 29, 2020

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Outline

Klein-Gordon chains Numerical integration

Taylor Series integrator Symplectic integrators

T-V decomposition MVD decomposition

Benchmarks

Time evolution

For low energy For high energy For medium energy

Energy localization versus initial conditions and chain length Interactive web application

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SLIDE 3

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Outline

Klein-Gordon chains Numerical integration

Taylor Series integrator Symplectic integrators

T-V decomposition MVD decomposition

Benchmarks

Time evolution

For low energy For high energy For medium energy

Energy localization versus initial conditions and chain length Interactive web application

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SLIDE 4

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Outline

Klein-Gordon chains Numerical integration

Taylor Series integrator Symplectic integrators

T-V decomposition MVD decomposition

Benchmarks

Time evolution

For low energy For high energy For medium energy

Energy localization versus initial conditions and chain length Interactive web application

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SLIDE 5

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Outline

Klein-Gordon chains Numerical integration

Taylor Series integrator Symplectic integrators

T-V decomposition MVD decomposition

Benchmarks

Time evolution

For low energy For high energy For medium energy

Energy localization versus initial conditions and chain length Interactive web application

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SLIDE 6

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Outline

Klein-Gordon chains Numerical integration

Taylor Series integrator Symplectic integrators

T-V decomposition MVD decomposition

Benchmarks

Time evolution

For low energy For high energy For medium energy

Energy localization versus initial conditions and chain length Interactive web application

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SLIDE 7

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Outline

Klein-Gordon chains Numerical integration

Taylor Series integrator Symplectic integrators

T-V decomposition MVD decomposition

Benchmarks

Time evolution

For low energy For high energy For medium energy

Energy localization versus initial conditions and chain length Interactive web application

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Klein-Gordon chains

H = H0 + ǫH1 =

∞

  • i=−∞

1 2p2

i + V (xi)

  • + 1

2ǫ

∞

  • i=−∞

(xi − xi−1)2 dpi dt = −∂H ∂xi , dxi dt = ∂H ∂pi ¨ xi = −V ′(xi) + ǫ(xi−1 − 2xi + xi+1).

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Klein-Gordon chains

V (x) = 1 2x2 + 1 4x4 H = H0 + ǫH1 =

∞

  • i=−∞

[1 2p2

i + 1

2x2

i + 1

4x4

i ] + 1

2ǫ

∞

  • i=−∞

(xi − xi−1)2 dpi dt = −∂H ∂xi , dxi dt = ∂H ∂pi ¨ xi = −xi − x3

i + ǫ(xi−1 − 2xi + xi+1)

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Numerical integration

From (xt0, pt0) to (xt1=t0+∆t, pt1=t0+∆t). Repeat.

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Taylor Series integrator

x (t1) = x (t0) + (t1 − t0) x′ (t0) + 1 2 (t1 − t0) 2x′′ (t0) + 1 6 (t1 − t0) 3x(3) (t0) + 1 24 (t1 − t0) 4x(4) (t0) + O

  • (t1 − t0) 5

p (t1) = p (t0) + (t1 − t0) p′ (t0) + 1 2 (t1 − t0) 2p′′ (t0) + 1 6 (t1 − t0) 3p(3) (t0) + 1 24 (t1 − t0) 4p(4) (t0) + O

  • (t1 − t0) 5

x′ = p p′ = −x − x3

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Taylor Series integrator

xi+1 = xi + τpi + 1 2τ 2 −x3

i − xi

  • + 1

6τ 3 −3pix2

i − pi

  • + 1

24τ 4 −6p2

i xi + x3 i − 3

  • −x3

i − xi

  • x2

i + xi

  • + O
  • τ 5

pi+1 = pi+τ

  • −x3

i − xi

  • +1

2τ 2 −3pix2

i − pi

  • +1

6τ 3 −6p2

i xi + x3 i − 3

  • −x3

i − xi

  • x2

i + xi

  • + 1

24τ 4 3pix2

i − 18pixi

  • −x3

i − xi

  • − 3x2

i

  • −3pix2

i − pi

  • − 6p3

i + pi

  • + O
  • τ 5
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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Taylor Series integrator

xi+1 = xi + τpi + 1 2τ 2 −x3

i − xi

  • + 1

6τ 3 −3pix2

i − pi

  • + 1

24τ 4 −6p2

i xi + x3 i − 3

  • −x3

i − xi

  • x2

i + xi

  • + O
  • τ 5

pi+1 = pi+τ

  • −x3

i − xi

  • +1

2τ 2 −3pix2

i − pi

  • +1

6τ 3 −6p2

i xi + x3 i − 3

  • −x3

i − xi

  • x2

i + xi

  • + 1

24τ 4 3pix2

i − 18pixi

  • −x3

i − xi

  • − 3x2

i

  • −3pix2

i − pi

  • − 6p3

i + pi

  • + O
  • τ 5
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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Taylor Series integrator

xi+1 = xi + τpi + 1 2τ 2 −x3

i − xi

  • + 1

6τ 3 −3pix2

i − pi

  • + 1

24τ 4 −6p2

i xi + x3 i − 3

  • −x3

i − xi

  • x2

i + xi

  • + O
  • τ 5

pi+1 = pi+τ

  • −x3

i − xi

  • +1

2τ 2 −3pix2

i − pi

  • +1

6τ 3 −6p2

i xi + x3 i − 3

  • −x3

i − xi

  • x2

i + xi

  • + 1

24τ 4 3pix2

i − 18pixi

  • −x3

i − xi

  • − 3x2

i

  • −3pix2

i − pi

  • − 6p3

i + pi

  • + O
  • τ 5
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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Taylor Series integrator

xi+1 = xi + τpi + 1 2τ 2 −x3

i − xi

  • + 1

6τ 3 −3pix2

i − pi

  • + 1

24τ 4 −6p2

i xi + x3 i − 3

  • −x3

i − xi

  • x2

i + xi

  • + O
  • τ 5

pi+1 = pi+τ

  • −x3

i − xi

  • +1

2τ 2 −3pix2

i − pi

  • +1

6τ 3 −6p2

i xi + x3 i − 3

  • −x3

i − xi

  • x2

i + xi

  • + 1

24τ 4 3pix2

i − 18pixi

  • −x3

i − xi

  • − 3x2

i

  • −3pix2

i − pi

  • − 6p3

i + pi

  • + O
  • τ 5
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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Taylor Series integrator

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Taylor Series integrator

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Symplectic integrators

Phase space volume is preserved: dpt0 ∧ dxt0 = dpt0+τ ∧ dxt0+τ A slightly perturbed Hamiltonian is preserved: ˜ H(xt0, pt0) = ˜ H(xt0+τ, pt0+τ) Example: Leapfrog integrator: pi+1/2 = pi + αi τ 2, xi+1 = xi + pi+1/2τ, pi+1 = pi+1/2 + αi+1 τ 2

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Symplectic integrators

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Symplectic integrators

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition

dp dt = −∂H ∂q , dq dt = ∂H ∂p HT = HT(p) = ⇒ ∂HT ∂q = 0 = ⇒ p(τ) = p(0), q(τ) = q(0) + τ ∂HT ∂p

  • p(0)

HV = HV (q) = ⇒ ∂HV ∂p = 0 = ⇒ q(τ) = q(0), p(τ) = p(0) − τ ∂HV ∂q

  • q(0)
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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition

H = T(p) + V (q) i = 1...k qi = qi−1 + τci ∂T ∂p

  • pi−1

pi = pi−1 + τdi ∂V ∂q

  • qi

→ q0 = q(0), p0 = p(0) qk = q(τ), pk = p(τ) ˜ H = H + O(τ n+1)

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition

n = 1 c1 1 d1 1 n = 2 c1 1/2 d1 1 c2 1/2 n = 4 c1, c4 1 2(2 − 21/3) d1, d3 1 2 − 21/3 c2, c3 1 − 21/3 2(2 − 21/3) d2 −21/3 2 − 21/3

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition

n = 1 c1 1 d1 1 n = 2 c1 1/2 d1 1 c2 1/2 n = 4 c1, c4 1 2(2 − 21/3) d1, d3 1 2 − 21/3 c2, c3 1 − 21/3 2(2 − 21/3) d2 −21/3 2 − 21/3

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition

n = 1 c1 1 d1 1 n = 2 c1 1/2 d1 1 c2 1/2 n = 4 c1, c4 1 2(2 − 21/3) d1, d3 1 2 − 21/3 c2, c3 1 − 21/3 2(2 − 21/3) d2 −21/3 2 − 21/3

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition

n = 1 c1 1 d1 1 n = 2 c1 1/2 d1 1 c2 1/2 n = 4 c1, c4 1 2(2 − 21/3) d1, d3 1 2 − 21/3 c2, c3 1 − 21/3 2(2 − 21/3) d2 −21/3 2 − 21/3

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Mixed variable decomposition (MVD)

H(q, p) = A(q, p) + ǫB(q, p) i = 1...k qi = fA(qi−1, pi−1) pi = fB(pi−1, qi) → q0 = q(0), p0 = p(0) qk = q(τ), pk = p(τ) ˜ H = H + O(τ 2nǫ + τ 2ǫ2)

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Mixed variable decomposition (MVD)

SABA1 c1, c2 1/2 d1 1 SABA2 c1, c3 1/2 − √ 3/6 d1, d2 1/2 c2 √ 3/3 SABA3 c1, c4 1/2 − √ 15/10 d1, d3 5/18 c2, c3 √ 15/10 d2 4/9

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Mixed variable decomposition (MVD)

SABA1 c1, c2 1/2 d1 1 SABA2 c1, c3 1/2 − √ 3/6 d1, d2 1/2 c2 √ 3/3 SABA3 c1, c4 1/2 − √ 15/10 d1, d3 5/18 c2, c3 √ 15/10 d2 4/9

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Mixed variable decomposition (MVD)

SABA1 c1, c2 1/2 d1 1 SABA2 c1, c3 1/2 − √ 3/6 d1, d2 1/2 c2 √ 3/3 SABA3 c1, c4 1/2 − √ 15/10 d1, d3 5/18 c2, c3 √ 15/10 d2 4/9

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Mixed variable decomposition (MVD)

SABA1 c1, c2 1/2 d1 1 SABA2 c1, c3 1/2 − √ 3/6 d1, d2 1/2 c2 √ 3/3 SABA3 c1, c4 1/2 − √ 15/10 d1, d3 5/18 c2, c3 √ 15/10 d2 4/9

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition and MVD in Klein-Gordon chain

T-V decomposition: H =

∞

  • i=−∞

[1 2p2

i + 1

2x2

i + 1

4x4

i ] + 1

2ǫ

∞

  • i=−∞

(xi − xi−1)2 ˙ xi = pi ˙ pi = − xi − x3

i + ǫ(xi−1 − 2xi + xi+1)

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition and MVD in Klein-Gordon chain

T-V decomposition: H =

∞

  • i=−∞

[1 2p2

i + 1

2x2

i + 1

4x4

i ] + 1

2ǫ

∞

  • i=−∞

(xi − xi−1)2 ˙ xi = pi ˙ pi = − xi − x3

i + ǫ(xi−1 − 2xi + xi+1)

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition and MVD in Klein-Gordon chain

T-V decomposition: H =

∞

  • i=−∞

[1 2p2

i + 1

2x2

i + 1

4x4

i ] + 1

2ǫ

∞

  • i=−∞

(xi − xi−1)2 ˙ xi = pi ˙ pi = − xi − x3

i + ǫ(xi−1 − 2xi + xi+1)

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition and MVD in Klein-Gordon chain

Mixed variable decomposition: H =

∞

  • i=−∞

[1 2p2

i + 1

2x2

i + 1

4x4

i ] + 1

2ǫ

∞

  • i=−∞

(xi − xi−1)2 ˙ xi = pi ˙ pi = − xi − x3

i + ǫ(xi−1 − 2xi + xi+1)

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SLIDE 36

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition and MVD in Klein-Gordon chain

Mixed variable decomposition: H =

∞

  • i=−∞

[1 2p2

i + 1

2x2

i + 1

4x4

i ] + 1

2ǫ

∞

  • i=−∞

(xi − xi−1)2 ˙ xi = pi ˙ pi = − xi − x3

i + ǫ(xi−1 − 2xi + xi+1)

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SLIDE 37

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

T-V decomposition and MVD in Klein-Gordon chain

Mixed variable decomposition: H =

∞

  • i=−∞

[1 2p2

i + 1

2x2

i + 1

4x4

i ] + 1

2ǫ

∞

  • i=−∞

(xi − xi−1)2 ˙ xi = pi ˙ pi = − xi − x3

i + ǫ(xi−1 − 2xi + xi+1)

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Single oscillator part - the hard way

¨ x = −x − x3 = ⇒ x(t) = c ∗ cn(λt − φ, k) p(t) = −cλ ∗ dn(λt − φ, k) ∗ sn(λt − φ, k) c =

  • −1 +

√ 1 + 4E E = x(0)2 2 + x(0)4 4 + p(0)2 2 λ =

  • 1 + c2

k =

  • c2

2λ φ = cn−1 x(0) c , k

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Single oscillator part - the smart way

¨ x = −x − x3 = ⇒ x(t) =

∞

  • k=0

∞

  • m=0

∞

  • j=0
  • ak,m,j ∗ xk

0 ∗ pm 0 ∗ tj

; p(t) =

∞

  • k=0

∞

  • m=0

∞

  • j=0
  • bk,m,j ∗ xk

0 ∗ pm 0 ∗ tj

; x0 ≡ x(0); p0 ≡ p(0); ak,m,j, bk,m,j ∈ R. x′ =

  • k
  • m

 xkpm

j

ak,m,j(ciτ)j   =

  • k
  • m
  • xkpm¯

ai,k,m

  • p′ =
  • k
  • m

 xkpm

j

bk,m,j(ciτ)j   =

  • k
  • m
  • xkpm¯

bi,k,m

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Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators

Single oscillator part - Factors for calculating x(t)

Factors p0

0 ≡ 1

p1 p2 p3 x0

0 ≡ 1

t − t3

6 + t5 120

− t5

20

x1 1 − t2

2 + t4 24

− t4

4

x2

t5 5 − t3 2

x3

t4 6 − t2 2

x4

9t5 40

x5

t4 8