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

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 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

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

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

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

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

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

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Klein-Gordon chains ∞ ∞ � 1 � + 1 � 2 p 2 � ( x i − x i − 1 ) 2 H = H 0 + ǫ H 1 = i + V ( x i ) 2 ǫ i = −∞ i = −∞ dp i dt = − ∂ H , dx i dt = ∂ H ∂ x i ∂ p i x i = − V ′ ( x i ) + ǫ ( x i − 1 − 2 x i + x i +1 ) . ¨

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Klein-Gordon chains V ( x ) = 1 2 x 2 + 1 4 x 4 ∞ ∞ [1 i + 1 i + 1 i ] + 1 � 2 p 2 2 x 2 4 x 4 � ( x i − x i − 1 ) 2 H = H 0 + ǫ H 1 = 2 ǫ i = −∞ i = −∞ dp i dt = − ∂ H , dx i dt = ∂ H ∂ x i ∂ p i x i = − x i − x 3 ¨ i + ǫ ( x i − 1 − 2 x i + x i +1 )

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Numerical integration From ( x t 0 , p t 0 ) to ( x t 1 = t 0 +∆ t , p t 1 = t 0 +∆ t ). Repeat.

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Taylor Series integrator x ( t 1 ) = x ( t 0 ) + ( t 1 − t 0 ) x ′ ( t 0 ) + 1 2 ( t 1 − t 0 ) 2 x ′′ ( t 0 ) + 1 6 ( t 1 − t 0 ) 3 x (3) ( t 0 ) + 1 24 ( t 1 − t 0 ) 4 x (4) ( t 0 ) + O ( t 1 − t 0 ) 5 � � p ( t 1 ) = p ( t 0 ) + ( t 1 − t 0 ) p ′ ( t 0 ) + 1 2 ( t 1 − t 0 ) 2 p ′′ ( t 0 ) + 1 6 ( t 1 − t 0 ) 3 p (3) ( t 0 ) + 1 24 ( t 1 − t 0 ) 4 p (4) ( t 0 ) + O ( t 1 − t 0 ) 5 � � x ′ = p p ′ = − x − x 3

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Taylor Series integrator x i +1 = x i + τ p i + 1 + 1 2 τ 2 � − x 3 6 τ 3 � − 3 p i x 2 � � i − x i i − p i + 1 24 τ 4 � − 6 p 2 i x i + x 3 − x 3 x 2 τ 5 � � � � � i − 3 i − x i i + x i + O +1 +1 − x 3 2 τ 2 � − 3 p i x 2 6 τ 3 � − 6 p 2 i x i + x 3 − x 3 x 2 � � � � � � p i +1 = p i + τ i − x i i − p i i − 3 i − x i i + x i + 1 24 τ 4 � 3 p i x 2 − x 3 − 3 x 2 − 3 p i x 2 − 6 p 3 τ 5 � � � � � � � i − 18 p i x i i − x i i − p i i + p i + O i

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Taylor Series integrator x i +1 = x i + τ p i + 1 + 1 2 τ 2 � − x 3 6 τ 3 � − 3 p i x 2 � � i − x i i − p i + 1 24 τ 4 � − 6 p 2 i x i + x 3 − x 3 x 2 τ 5 � � � � � i − 3 i − x i i + x i + O +1 +1 − x 3 2 τ 2 � − 3 p i x 2 6 τ 3 � − 6 p 2 i x i + x 3 − x 3 x 2 � � � � � � p i +1 = p i + τ i − x i i − p i i − 3 i − x i i + x i + 1 24 τ 4 � 3 p i x 2 − x 3 − 3 x 2 − 3 p i x 2 − 6 p 3 τ 5 � � � � � � � i − 18 p i x i i − x i i − p i i + p i + O i

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Taylor Series integrator x i +1 = x i + τ p i + 1 + 1 2 τ 2 � − x 3 6 τ 3 � − 3 p i x 2 � � i − x i i − p i + 1 24 τ 4 � − 6 p 2 i x i + x 3 − x 3 x 2 τ 5 � � � � � i − 3 i − x i i + x i + O +1 +1 − x 3 2 τ 2 � − 3 p i x 2 6 τ 3 � − 6 p 2 i x i + x 3 − x 3 x 2 � � � � � � p i +1 = p i + τ i − x i i − p i i − 3 i − x i i + x i + 1 24 τ 4 � 3 p i x 2 − x 3 − 3 x 2 − 3 p i x 2 − 6 p 3 τ 5 � � � � � � � i − 18 p i x i i − x i i − p i i + p i + O i

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Taylor Series integrator x i +1 = x i + τ p i + 1 + 1 2 τ 2 � − x 3 6 τ 3 � − 3 p i x 2 � � i − x i i − p i + 1 24 τ 4 � − 6 p 2 i x i + x 3 − x 3 x 2 τ 5 � � � � � i − 3 i − x i i + x i + O +1 +1 − x 3 2 τ 2 � − 3 p i x 2 6 τ 3 � − 6 p 2 i x i + x 3 − x 3 x 2 � � � � � � p i +1 = p i + τ i − x i i − p i i − 3 i − x i i + x i + 1 24 τ 4 � 3 p i x 2 − x 3 − 3 x 2 − 3 p i x 2 − 6 p 3 τ 5 � � � � � � � i − 18 p i x i i − x i i − p i i + p i + O i

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Taylor Series integrator

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Taylor Series integrator

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Symplectic integrators Phase space volume is preserved: d p t 0 ∧ d x t 0 = d p t 0 + τ ∧ d x t 0 + τ A slightly perturbed Hamiltonian is preserved: H ( x t 0 , p t 0 ) = ˜ ˜ H ( x t 0 + τ , p t 0 + τ ) Example: Leapfrog integrator: τ p i +1 / 2 = p i + α i 2 , x i +1 = x i + p i +1 / 2 τ, τ p i +1 = p i +1 / 2 + α i +1 2

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Symplectic integrators

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators Symplectic integrators

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators T-V decomposition dt = − ∂ H dt = ∂ H dp dq ∂ q , ∂ p � ⇒ ∂ H T q ( τ ) = q (0) + τ ∂ H T � H T = H T ( p ) = = 0 = ⇒ p ( τ ) = p (0) , � ∂ q ∂ p � p (0) � ⇒ ∂ H V p ( τ ) = p (0) − τ ∂ H V � H V = H V ( q ) = = 0 = ⇒ q ( τ ) = q (0) , � ∂ p ∂ q � q (0)

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators T-V decomposition H = T ( p ) + V ( q ) i = 1 ... k q 0 = q (0) , p 0 = p (0) � ∂ T � q i = q i − 1 + τ c i � ∂ p → � p i − 1 q k = q ( τ ) , p k = p ( τ ) � ∂ V � p i = p i − 1 + τ d i � ∂ q � q i H = H + O ( τ n +1 ) ˜

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators T-V decomposition n = 1 c 1 1 d 1 1 n = 2 c 1 1 / 2 d 1 1 1 / 2 c 2 n = 4 1 1 c 1 , c 4 d 1 , d 3 2(2 − 2 1 / 3 ) 2 − 2 1 / 3 1 − 2 1 / 3 − 2 1 / 3 c 2 , c 3 d 2 2(2 − 2 1 / 3 ) 2 − 2 1 / 3

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators T-V decomposition n = 1 c 1 1 d 1 1 n = 2 c 1 1 / 2 d 1 1 1 / 2 c 2 n = 4 1 1 c 1 , c 4 d 1 , d 3 2(2 − 2 1 / 3 ) 2 − 2 1 / 3 1 − 2 1 / 3 − 2 1 / 3 c 2 , c 3 d 2 2(2 − 2 1 / 3 ) 2 − 2 1 / 3

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators T-V decomposition n = 1 c 1 1 d 1 1 n = 2 c 1 1 / 2 d 1 1 1 / 2 c 2 n = 4 1 1 c 1 , c 4 d 1 , d 3 2(2 − 2 1 / 3 ) 2 − 2 1 / 3 1 − 2 1 / 3 − 2 1 / 3 c 2 , c 3 d 2 2(2 − 2 1 / 3 ) 2 − 2 1 / 3

Outline Klein-Gordon chains Taylor Series integrator Symplectic integrators T-V decomposition n = 1 c 1 1 d 1 1 n = 2 c 1 1 / 2 d 1 1 1 / 2 c 2 n = 4 1 1 c 1 , c 4 d 1 , d 3 2(2 − 2 1 / 3 ) 2 − 2 1 / 3 1 − 2 1 / 3 − 2 1 / 3 c 2 , c 3 d 2 2(2 − 2 1 / 3 ) 2 − 2 1 / 3