joint works with prof.dr.bülent karasözen ayhan aydın...

IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY

Geometric Integration:Symplectic and multisymplectic methods

joint works with Prof.Dr.Bülent Karasözen

Ayhan AydınAtılım University Mathematics Department, Ankara

24.11.2017

Numerical Analysis and Scientific Computing:Workshop in Honour of Bülent Karasözen’s 67th Birthday

Middle East Technical University, Ankara-Turkey


M.Sc. Thesis

Poisson Integrators for Completely Integrable Hamiltonian Systems,Supervisor: Prof.Dr.Bülent Karasözen, 1998

Ph.D. Thesis

Geometric Integrators for the Coupled Nonlinear Schrödinger Equations,Supervisor: Prof.Dr.Bülent Karasözen, 2005

B.Karasözen, A. Aydin, Chapter 3 - Multisymplectic Integrators forCoupled Nonlinear Partial Differential Equations (pp.267-296),Series: Computer Science and Robotics, Book: Computer Physics,Editors: Brian S. Doherty and Amy N. Molloy, Nova Science Publishers(2012)


PAPERS

[1] A. Aydın, B. Karasözen, (2007) Symplectic and multi-symplecticmethods for coupled nonlinear Schrödinger equations with periodicsolutions, Computer Physics Communications 177, 566-583.[2] A. Aydın, B. Karasözen, (2008) Symplectic and multi-symplecticLobatto methods for the "good" Boussinesq equation, Journal ofMathematical Physics 49, 083509.[3] A. Aydın, B. Karasözen. (2008) Multisymplectic schemes for thecomplex modified Korteweg-de Vries equation AIP ConferenceProceedings 1048, 60.[4] A. Aydın, B. Karasözen, (2009) Multi-symplectic integration ofcoupled nonlinear Schrödinger system with soliton solutions,International Journal of Computer Mathematics 86, 864-882.[5] A. Aydın, B. Karasözen. (2010) Multisymplectic box schemes forthe complex modified Korteweg-de Vries equation. Journal ofMathematical Physics 51:8, 083511.[6] A.Aydın, B.Karasözen, (2011) Lobatto IIIA-IIIB Discretization ofthe strongly coupled nonlinear Schrödinger equation, Journal ofComputational and Applied Mathematics, 235, 4770-4779.


GEOMETRIC INTEGRATION

Simple Harmonic oscillator

md2x

dt2+ kx = 0

k = m = 1 :dx

dt= p,

dp

dt= −x

(1)


Euler method

dy

dt= f (y), yn+1 = yn + h f (yn) (2)

Euler method for Hamiltonian system (1)

xn+1 = xn + h pn

pn+1 = pn − h xn

(3)

Implicit midpoint rule

dy

dt= f (y), yn+1 = yn + h f

(yn+1 + yn

2

)(4)

Implicit midpoint rule for Hamiltonian system (1)

xn+1 = xn + h(

pn+1+pn

2

)

pn+1 = pn − h(

xn+1+xn

2

) (5)


−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3Forward Euler

p

x

I.C.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5Implicit Midpoint rule

p

x

I.C.

0 5 10 15−3

−2

−1

0

1

2

3

t

x

0 5 10 15−1.5

−1

−0.5

0

0.5

1

1.5

t

x

Not all schemes can give reliable numerical results. Inappropriatediscretization may induce unphysical "blow-up" and "numericalchaos" ( Fei, et al. Numerical Simulation of NLS: A New Conservative Scheme )


REASON? ***CONSERVATION OF ENERGY***

The equation of the simple harmonic oscillator is aHamiltonian system with the Hamiltonian

H(x, p) =p2

2+

x2

2(6)

Hamilton’s equation of motion

dx

dt= Hp = p,

dp

dt= −Hx = −x (7)

dH

dt=

∂H

∂x

(dx

dt

)+∂H

∂p

(dp

dt

)= 0 (8)

Preservation of circle ! p2 + x2 = C2


Euler method for Hamiltonian system (1)

xn+1 = xn + h pn

pn+1 = pn − h xn

(9)

x2n+1 + p2

n+1 = (1 + h2)(

x2n + p2

n

)= (1 + h2)C2 (10)

i.e. the area enclosed by the discrete solution (xn, pn)T has

increased by a factor of 1 + h2.

Implicit midpoint rule for Hamiltonian system (1)

xn+1 = xn + h(

pn+1+pn

2

)

pn+1 = pn − h(

xn+1+xn

2

) (11)

x2n+1 + p2

n+1 = x2n + p2

n = · · · = x20 + p2

0 (12)


GEOMETRIC INTEGRATION

Designing numerical methods that share qualitative features ofthe differential equation

Symplectic and multisymplectic methods for DEs withHamiltonian structure (Newton, Stormer, Verlet, deVogelaerre, Feng Kang, Sanz-Serna, Scovel, Hairer &Lubich, Bennetin & Gorgili, . . . )

Volume and energy conservation in DEs (Feng Kang,McLachlan & Quispel, . . . )

Methods respecting Poission and Lie-Poisson structure(Marsden, Lewis & Simo, Ratiu, . . . )

...


HAMILTONIAN EQUATION OF MOTION

H(q, p) : Rn × R

n → R

dqi

dt=∂H

∂pi,

dpi

dt= −

∂H

∂qi, (13)

or

dz

dt=

(∂H∂pi

−∂H∂qi

)= J ∇H(z),

(qT

pT

), J =

(0n In

−In 0n

)(14)

i = 1, 2, · · · ,n. The exact solution of the Hamiltonian system(13) has the following properties

Energy conservation

H(q(0), p(0)) = H(q(t), p(t)) (15)


The flow map φt,H(z), i.e.H(t, z0) = φt,H(z0) is symplectic, i.e.

(∂

∂zφt,H(z)

)T

J−1

(∂

∂zφt,H(z)

)= J−1 (16)

ord∑

i=1

dp(0) ∧ dq(0) =d∑

i=1

dp(t) ∧ dq(t) (17)

Preservation of the symplectic structure: ddtω = 0

ω = dp ∧ dq = dptdq − dqtdp


CONSERVATIVE NUMERICAL METHOD

Definition

A one-step method is called energy conserving, as applied tothe Hamiltonian system (13) generating solution (qn+1, pn+1)

H(qn+1, pn+1) = H(qn, pn) (18)

Definition

A one-step method is called symplectic, as applied to theHamiltonian system (13) generating solution (qn+1, pn+1)

(∂(qn+1, pn+1)

∂(qn, pn)

)T

J−1

(∂(qn+1, pn+1)

∂(qn, pn)

)= J−1 (19)


SOME SYMPLECTIC METHOD

Symplectic Euler

q = Hp(q, p), p = −Hq(q, p)

qn+1 = qn + hHp(qn, pn+1)

pn+1 = pn − hHq(qn, pn+1)

orqn+1 = qn + hHp(q

n+1, pn)pn+1 = pn − hHq(q

n+1, pn)(20)

Implicit midpoint rule

dz

dt= J−1∇H(z)

zn+1 = zn + hJ−1∇H

(zn+1 + zn

2

)(21)


Strömer-Verlet scheme

q = Hp(q, p), p = −Hq(q, p)

qn+1/2 = qn + h2 Hq(q

n+1/2, pn)

pn+1 = pn − h2

(Hp(q

n+1/2, pn) + Hp(qn+1/2, pn+1)

)

qn+1 = qn+1/2 + h2 Hq(q

n+1/2, pn+1)

(22)

Symplectic Runge-Kutta methods

The s–stage Runge–Kutta (RK) method

c A

bT

withbiaij + bjaji − bibj = 0, for all i, j = 1, 2, · · · , s (23)


ci aij

bi

ci aij

bi

Partitioned Runge-Kutta methods

The s-stage PRK method for the ODEq = f (q, p), p = g(q, p)

Qi = qn + ∆t∑s

j=1 aijf (Qj,Pj), i = 1, · · · , s

Pi = pn + ∆t∑s

j=1 aijg(Qj,Pj), i = 1, · · · , s

qn+1 = qn + ∆t∑s

j=1 bjf (Qj,Pj), n = 0, 1, · · ·

pn+1 = pn + ∆t∑s

j=1 bjg(Qj,Pj), n = 0, 1, · · ·

Symplecticity condition for PRK

biaij + bjaji − bibj = 0,

bi = bi, i, j = 1, · · · , s


THE SECOND ORDER 2-STAGE SYMPLECTIC LOBATTO

IIIA-IIIB METHOD

q = Hp(q, p), p = −Hq(q, p)

IIIA:0 0 01 1/2 1/2

1/2 1/2, IIIB:

0 1/2 01 1/2 0

1/2 1/2.


Multisymplectic Hamiltonian PDE Systems

Kzt + Lzx = ∇zS(z), z ∈ Rn (24)

Preservation of the multisymplectic structure

ωt + κx = 0 with ω := 12 dz ∧ Kdz, κ := 1

2 dz ∧ Ldz (25)

If S does not depend on (t, x) then (1) has

Local energy and momentum conservation lawsEt + Fx = 0, E(z) = S(z) − 1

2 zTLzx, F(z) = 12 zTLzt

It + Gx = 0, G(z) = S(z) − 12 zTKzt, F(z) = 1

2 zTKzx

Integrating local conservation laws over the spatial domainwith periodic boundary conditions yields

Global energy and momentum conservation lawsd

dt

∫ L

0E(z) dx = 0,

d

dt

∫ L

0I(z) dx = 0


Multisymplectic integrators: Kzt + Lzx = ∇zS(z),

Preserve the multisymplectic structure: ωt + κx = 0, i.e.

∂m,nt ωn

m + ∂m,nx κn

m = 0 (26)

Marsden, Patrick, and Sckoller (1998) - methods thatapproximate the Lagrangian by a sum and take variations

Bridges and Reich (2001) - methods that discretise themultisymplectic pdes and preserve a MSCL


SOME MULTISYMPLECTIC SCHEMES

Introduce

Dxψnm =

ψnm+1 − ψn

m

∆x, Mxψ

nm =

ψnm+1 + ψn

m

2

Dtψnm =

ψn+1m − ψn

m

∆t, Mtψ

nm =

ψn+1m + ψn

m

2

The Preissman box scheme

KDxMtznm + LDtMxzn

m = ∇S(MtMxznm) (27)

with discrete multisymplectic conservation law

dz ∧ KDxMtdznm + dzn

m ∧ LDxMtdznm = 0.


A non-unique splitting of the matrices K = K+ + K− andL = L+ + L− with KT

+ = −K− and LT+ = −L−

The Euler box scheme

K+δ+t zn

m + K−δ−t zn

m + L+δ+x zn

m + L−δ−x zn

m = ∇zS(znm) (28)

which satisfies the discrete multisymplectic conservation laws

δ+t ωnm + δ+x κ

nm = 0 (29)

where ωnm = dzn−1

m ∧ K+dznm and κn

m = dznm−1 ∧ L+dzn

m.

Here δ+t , δ+x and δ−t , δ

−x represents forward and backward finite

difference operators for first-order time and space derivativediscretization, respectively.


1 Multisymplectic box scheme has remarkable energy and momentumconservation in long time integration

2 If S(z) is a quadratic function of z, then both discrete energy anddiscrete momentum are conserved to machine accuracy

3 Modified equation is also multisymplectic by Backward Error Analysis:Instead of asking "what is the numerical error for our problem", it isasked "which nearby problem is solved exactly by our method?".

4 Modified LECL and LMCL satisfies 4th order accuracy (NLS).

5 For nonlinear Hamiltonian PDEs, the local energy conservation law(LECL) and local momentum conservation law (LMCL) will not bepreserved exactly.

6 Recently, local energy and local momentum preserving multisymplecticmethods based on AVF method are proposed.


Average Vector Field(AVF) Method

For ODE y′ = f (y), y ∈ Rd, the AVF method is the map

yn → yn+1 defined by

yn+1 − yn

∆t=

∫ 1

0f ((1 − ξ)yn + ξyn+1)dξ (30)


Local energy-preserving MS method

Space : Implicit midpoint, Time : AVF

KDtMxznm+LDxMtzn

m =

∫ 1

0∇zS((1−ξ)Mxzn

m+ξMxzn+1m )dξ (31)

which satisfies the LECL:

DtMxEnm + DxMtF

nm = 0 (32)

Local momentum-preserving MS method

Space : AVF, Time : Implicit midpoint

KDtMxznm +LDxMtzn

m =

∫ 1

0∇zS((1− ξ)Atzn

m + ξAtzn+1m )dξ (33)

which satisfies the LMCL:

DtMxInm + DxMtG

nm = 0 (34)


DISPERSION

The fundamental idea

When a PDE is linear and constant coefficients, it admits "planewave" solution of the form

u(x, t) = ei(ξx−ωt), ξ ∈ R, ω ∈ C, (35)

where ξ is the wave numeber and ω is the frequency.

u(x, t) represents a sinusoidal wave of length 2π/ξ, period2π/ω.

Dispersion Relation = ω = ω(ξ)

Group velocity =dω

dξ

(36)


EXAMPLE(1)

The advection equation

ut + ux = 0, xL ≤ x ≤ xR, 0 ≤ t ≤ 2

u(x, 0) = e−16x(x−1/2)2 sin(ξx)

u(xL, t) = u(xR, t)

(37)

Dispersion relation : ω = ξ

Group velocity :dω

dξ= 1

Leap–frog

Un+1j = Un−1

j − λ(Unj+1 − Un

j−1), λ =∆t

∆x(38)

11Lloyd N. Trefethen, Finite Difference and Spectral methods for

ordinary and partial differential equations, Cornell University, (1996).


Discrete plane wave

U(xj, tn) = ei(ξxj−ωtn) = ei(ξj∆x−ωn∆t) (39)

Aliasing

eiξxj = ei(ξ+ 2π

∆x)xj

v(ξ) = eiξxj , ξ ∈ [−π/∆x, π/∆x]

(40)


Numeric dispersion relation

sin(ω∆t) = λ sin(ξ∆x) (41)

where the fundamental domain is(ξ, ω) = [−π/∆x, π/∆x] × [−π/∆t, π/∆t]

Numeric group velocity

dω

dξ=

cos(ξ∆x)

cos(ω∆t)(42)


0 0.5 1 1.5 2 2.5 3−1

−0.5

0

0.5

1

x−axis

u−a

xis

t=0

(a) The initial wave packet

0 0.5 1 1.5 2 2.5 3−1

−0.5

0

0.5

1

x−axis

u−a

xis

t=2

(b) After the wave propagate to t = 2 by LF

Figure: Propagation of wave packet of ut + ux = 0 by LF with λ = 0.4.


Recall:

Dispersion relation : ω = ξ

Group velocity :dω

dξ= 1

Position of wave

velocity =Distance

Time

1 =Distance

2

Distance = 2

Position of the wave should be at x = 0.5 + 2 = 2.5.However, it is NOT!!!

Question

What is the problem in the evolution of the wave packet under LFwith λ = 0.4?


Answer

The error comes from the numerical group velocity under LF

Velocity from the figure

Position of the wave packet:(t0, x0) = (0, 0.5) (tfin, xfin) ≈ (2, 1.97)

Velocity (by LF) =Distance

Time

=1.97 − 0.5

2

= 0.7350


Group Velocity from dispersion relation

Recall:dω

dξ= 1

Numerical Group Velocity from numeric dispersion relation

Recall:dω

dξ=

cos(ξ∆x)

cos(ω∆t)

We choose ξ so that there are 8 grid points per wavelength:ξ∆x = 2π/8

ξ ≈ 125.7, ∆x = 1/160, ,∆t = 0.0025

dω

dξ= 0.7372

One can determine the location of the wave by using thenumerical group velocity.


u(x, t) = uei(ξx−ω(ξ)t), (43)

where u is constant.

For ω = Re(ω) + iIm(ω) = ωr + iωι

u(x, t) = uei(ξx+ωrt)e−ωιt. (44)

ωι > 0: the plane wave will decay,

ωι < 0: the plane wave will grow without a bound.

ωι = 0: the plane wave will neither grow nor decay.


Classification

If the plane wave do not grow with time and at least onemode(or wave) decays, then the PDE is said to be dissipative.

If the plane wave neither decay nor grow, then the PDE is callednon–dissipative.

The PDE is called dispersive when the plane wave of differentwave lengths (or wave numbers) propagate at different speeds.

Classification

In practice,PDE containing only

even ordered x derivatives are dissipative.(Eg. NLS:ut + uxx + |u|2u = 0)

odd ordered x derivatives is non-dissipative and when theorder is greater than one the PDE is dispersive.(Eg. KdV: ut + uux + uxxx = 0)


"Preservation of the dissipative or dispersive properties of the PDE bynumerical scheme"

i.e.

"Find a numerical method that has the same dissipative or dispersiveproperties with the corresponding PDE "


The two coupled nonlinear Schrödinger (CNLS) equations aregiven by

i∂ψ1

∂t+ α1

∂2ψ1

∂x2+ (σ1 |ψ1|

2 + v12 |ψ2|2)ψ1 = 0

i∂ψ2

∂t+ α2

∂2ψ2

∂x2+ (σ2 |ψ2|

2 + v21 |ψ1|2)ψ2 = 0

(45)

By decomposing the complex functions ψ1, ψ2

ψ1(x, t) = q1(x, t) + iq2(x, t), ψ2(x, t) = q3(x, t) + iq4(x, t)


These equations represent an infinite-dimensional Hamiltoniansystem in the phase space z = (q1, q2, q3, q4)

T

zt = J−1 δH

δz, J =

0 −1 0 01 0 0 00 0 0 −10 0 1 0

(46)

where the Hamiltonian is

H(z) =

∫ {W −

α1

2

((∂q1

∂x)2 + (

∂q2

∂x)2

)−α2

2

((∂q3

∂x)2 + (

∂q4

∂x)2

)}dx,

(47)

with W =1

4(σ1(q

21 + q2

2)2 + σ2(q

23 + q2

4)2) +

v

2(q2

1 + q22)(q

23 + q2

4).


SEMI-IMPLICIT SYMPLECTIC SCHEME

Hamiltonian splitting: H = HLin + HNon.

HLin = −

∫ {α1

2

((∂q1

∂x)2 + (

∂q2

∂x)2

)+α2

2

((∂q3

∂x)2 + (

∂q4

∂x)2

)}dx,

(48)and

HNon =

∫ {1

4(σ1(q

21 + q2

2)2 + σ2(q

23 + q2

4)2) +

v

2(q2

1 + q22)(q

23 + q2

4)

}dx.

(49)


The linear vector field :

∂q1

∂t= −α1

∂2q2

∂x2,

∂q2

∂t= α1

∂2q1

∂x2,

∂q3

∂t= −α2

∂2q4

∂x2,

∂q4

∂t= α2

∂2q3

∂x2

(50)

and the nonlinear vector field :

∂q1

∂t= −(σ1(q

21 + q2

2) + v(q23 + q2

4))q2,

∂q2

∂t= (σ1(q

21 + q2

2) + v(q23 + q2

4))q1,

∂q3

∂t= −(v(q2

1 + q22) + σ2(q

23 + q2

4))q4,

∂q4

∂t= (v(q2

1 + q22) + σ2(q

23 + q2

4))q3.

(51)


DISCRETIZATION

Linear part: H = Hlin(Z) Discretization in space using thecentral difference approximation for the second orderderivatives, we get the semi-discretized linear subproblem

dZ

dt= J−1∇Hlin(Z), with Z := (Q1,Q2,Q3,Q4)

T (52)

where Qi := (qi1, qi2, · · · , qiN), i = 1, · · · , 4 and

J :=

0 −I 0 0I 0 0 00 0 0 −I0 0 I 0

where I is the N × N identity matrix. Time discretization:Symplictic implicit mid-point rule to the linear part (52)

Zn+1 − Zn

∆t= J−1∇Hlin(

Zn+1 + Zn

2), (53)


The semi-discrete nonlinear subproblem : H = Hnon(Z)

∂q1m

∂t= −

[σ1

(q1

2m + q2m

2)

+ v(q3

2m + q4

2m

) ]q2m,

∂q2m

∂t=

[σ1

(q1

2m + q2m

2)

+ v(q3

2m + q4

2m

) ]q1m,

∂q3m

∂t= −

[v(q1

2m + q2m

2)

+ σ2

(q3

2m + q4

2m

) ]q4m,

∂q4m

∂t=

[v(q1

2m + q2m

2)

+ σ2

(q3

2m + q4

2m

) ]q3m.

(54)

Time discretization: Symplictic implicit mid-point rule to thenon-linear part.


COMPOSITION

The solutions of linear and nonlinear subproblem arecomposed by the second order symmetric integrator

ϕ2(∆t) = e∆t2

Hnon ◦ e∆tHlin ◦ e∆t2

Hnon (55)

which results a symplectic integrator for the CNLS system (60).


MULTI-SYMPLECTIC STRUCTURE OF CNLS EQUATION

Mzt + Kzx = ∇zS(z), z ∈ Rn (56)

Introducing

p1 = α1∂q1

∂x, p2 = α1

∂q2

∂x, p3 = α2

∂q3

∂x, p4 = α2

∂q4

∂x, (57)

and z = (q1, q2, q3, q4, p1, p2, p3, p4)T

M =

(−J 0

0 0

), K =

(0 −II 0

), J =

0 −1 0 01 0 0 00 0 0 −10 0 1 0

(58)where S(z) = W + 1

2α1

(p2

1 + p22

)+ 1

2α2

(p2

3 + p24

)with

W = σ14

(q2

1 + q22

)2+ σ2

4

(q2

3 + q24

)2+ v

2

(q2

1 + q22

) (q2

3 + q24

), 0, I

denote the 4 × 4 zero and identity matrices respectively.


PREISSMAN SCHEME

Mzn+1

m+1/2 − znm+1/2

∆t+ K

zn+1/2m+1 − z

n+1/2m

∆x= ∇zS(z

n+1/2m+1/2),

with

znm+1/2 =

znm + zn

m+1

2, z

n+1/2m =

znm + zn+1

m

2,

z = zn+1/2m+1/2 =

znm + zn

m+1 + zn+1m + zn+1

m+1

4

Local energy and momentum conservation laws when S isindependent of x and t

Et + Fx = 0, E(z) = S(z) − 12 zTLzx, F(z) = 1

2 zTLzt,

It + Gx = 0, G(z) = S(z) − 12 zTMzt, I(z) = 1

2 zTMzx.(59)


Destable initial conditions: ψ1(x, 0) =0.5(1 − 0.1 cos(x/2)), ψ2(x, 0) = 0.5(1 − 0.1 cos((x + θ)/2))


The coupled nonlinear Schrödinger (CNLS) system

i

(∂ψ1

∂t+ δ

∂ψ1

∂x

)+ α

∂2ψ1

∂x2+ (|ψ1|

2 + e |ψ2|2)ψ1 = 0

i

(∂ψ2

∂t− δ

∂ψ2

∂x

)+ α

∂2ψ2

∂x2+ (e |ψ2|

2 + |ψ1|2)ψ2 = 0

(60)

Introducing

p1 + ip2 = α∂ψ1

∂x+

1

2iδψ1, p3 + ip4 = α

∂ψ2

∂x−

1

2iδψ2 (61)

with the state variable z = (q1, q2, q3, q4, p1, p2, p3, p4)T


M =

(J 00 0

), L =

(0 −II 0

), (62)

S(z) =1

2α(p2

1 + p22 + p2

3 + p24) +

δ

2α(p1q2 − p2q1 − p3q4 + p4q3)

+δ2

8α(q2

1 + q22 − q2

3 − q24) +

1

4(q2

1 + q22)

2

+1

4(q2

3 + q24)

2 +e

2(q2

1 + q21)(q

23 + q2

4)

J =

0 1 0 0−1 0 0 00 0 0 10 0 −1 0

Et + Fx = 0, E(z) = S(z) − 12 zTLzx, F(z) = 1

2 zTLzt,

It + Gx = 0, G(z) = S(z) − 12 zTMzt, I(z) = 1

2 zTMzx.(63)


The Preissman scheme

Mzn+1

m+1/2 − znm+1/2

∆t+ L

zn+1/2m+1 − z

n+1/2m

∆x= ∇zS(z

n+1/2m+1/2),

with

znm+1/2 =

znm + zn

m+1

2, z

n+1/2m =

znm + zn+1

m

2,

zn+1/2m+1/2 =

znm + zn

m+1 + zn+1m + zn+1

m+1

4


pj can be eliminated and one obtains a new scheme. Themulti-symplectic six-point scheme can be written explicitly as

i

[(ψ1

n+1m−1+2ψ1

n+1m +ψ1

n+1m+1)−(ψ1

nm−1+2ψ1

nm+ψ1

nm+1)

4∆t

]

−i

[δ

(ψ1n+1m−1−ψ1

n+1m+1)+(ψ1

nm−1−ψ1

nm+1)

4∆x

]

+α(ψ1

n+1m−1−2ψ1

n+1m +ψ1

n+1m+1)+(ψ1

nm−1−2ψ1

nm+ψ1

nm+1)

2∆x2 + κ1ψ1 + κ1ψ1 = 0,

i

[(ψ2

n+1m−1+2ψ2

n+1m +ψ2

n+1m+1)−(ψ2

nm−1+2ψ2

nm+ψ2

nm+1)

4∆t

]

+i

[δ

(ψ2n+1m−1−ψ2

n+1m+1)+(ψ2

nm−1−ψ2

nm+1)

4∆x

]

+α(ψ2

n+1m−1−2ψ2

n+1m +ψ2

n+1m+1)+(ψ2

nm−1−2ψ2

nm+ψ2

nm+1)

2∆x2 + κ2ψ2 + κ2ψ2 = 0

where κ1 = |ψ1|2 + e|ψ2|

2, κ1 = |ψ1|2 + e|ψ2|

2, κ2 = e|ψ1|2 + |ψ2|

2,

κ2 = e|ψ1|2 + |ψ2|

2 with ψ = ψn+1/2m−1/2 and ψ = ψ

n+1/2m+1/2.


Dispersion relations of the Preissman scheme and the sixpoint scheme for the CNLS equation are discussed


International Conference on Numerical Analysis and AppliedMathematics(ICNAAM2008),

Kos-Greek(2008)


The complex modified Korteweg-de Vries (CMKdV) equation

∂ψ

∂ψ+∂3ψ

∂x3+ α

∂(|ψ|2ψ

)

∂x= 0, −∞ < x <∞, t > 0, (64)

where ψ(x, t) is a complex-valued function. By decomposingψ(x, t) = u(x, t) + iv(x, t), i2 = −1 and introducingη1, η2, φ1, φ2,w1,w2 with wx(x, t) = η1(x, t) + iη2(x, t),− 1

2 w(x, t) = φ1x(x, t) + iφ2x(x, t),12 w1(x, t) = −φ1t(x, t) + η1x(x, t) + α(u2 + v2)u,12 w2(x, t) = −φ2t(x, t) + η2x(x, t) + α(u2 + v2)v, the CMKdVequation can be rewritten as

Kzt + Lzx = ∇zS(z) (65)

with state variable z = (u, v, φ1, φ2, η1, η2,w1,w2)T and two

skew-symmetric matrices


K =

(J1 04

04 04

), L =

(04 I4

−I4 04

), with J1 =

(02 −I2

I2 02

)

The first-order system (65) has the multisymplecticconservation law

ωt + κx = 0 with ω = 12 dz ∧ Kdz and κ = 1

2 dz ∧ Ldz.(66)

Multisymplectic splitting

L =∑N

j=1 L(j) and S(z) =∑N

j=1 S(j)(z)

Kzt + L(j)zx = ∇zS(j)(z). (67)

ωt + κ(j)x = 0, (68)

where κ(j) = 1/2L(j)dz ∧ dz.


L = L(1) + L(2)

L(1) =

(04 J2

−J2 04

), L(2) =

(04 J3

−J3 04

),

J2 =

(I2 02

0212 I2

), J3 =

(02 02

0212 I2

),

S(z) = S(1) + S(2)

S(1) =1

4(uw1+vw2)−

1

2(η2

1+η22),S

(2) =1

4(uw1+vw2)−

α

4(u2+v2)2.

Kzt + L(1)zx = ∇zS(1)(z) gives wt + wxxx = 0 Lin. eq. (69)

Kzt + L(2)zx = ∇zS(2)(z) gives wt +α(|w|2w

)

x= 0 Nonlin. eq.

(70)


Preissman scheme for Kzt + L(j)zx = ∇zS(j)(z),j = 1, 2

(awn+1m + 3bwn+1

m+1 + 3awn+1m+2 + bwn+1

m+3)

−(bwnm + 3awn

m+1 + 3bwnm+2 + awn

m+3) = 0(71)

wn+1m+1 + wn+1

m − wnm+1 − wn

m + e[(wn+1m+1 + wn

m+1)|wn+1m+1 + wn

m+1|2

−(wn+1m + wn

m)|wn+1m + wn

m|2] = 0

(72)

The splitting is advanced in time according to the second–ordercompositions

exp

(∆t

2L

)exp (∆tN ) exp

(∆t

2L

). (73)


The complex modified Korteweg-de Vries (CMKdV) equation

∂ψ

∂ψ+∂3ψ

∂x3+ α

∂(|ψ|2ψ

)

∂x= 0, −∞ < x <∞, t > 0, (74)

where ψ(x, t) is a complex-valued function. Introduce

Dxψnm =

ψnm+1 − ψn

m

∆x, Mxψ

nm =

ψnm+1 + ψn

m

2

Dtψnm =

ψn+1m − ψn

m

∆t, Mtψ

nm =

ψn+1m + ψn

m

2

The Preissman scheme for the CMKDV eq.

KDxMtz + LDtMxz = ∇S(MtMxz) (75)

with discrete multisymplectic conservation law

dz ∧ KDxMtdz + dz ∧ LDxMtdz = 0.


The 12-point scheme for CMKDV

Eliminating the auxiliary variables φ1,φ2,η1,η2,w1,w2 yields

DtMtM3xψ + D3

xM2tψ + αDxMtMx(|MtMxψ|MtMxψ) = 0. (76)

Expressed using finite difference stencils, the 12-point scheme(76) is written as

1

16∆t

1 3 3 10 0 0 0−1 −3 −3 −1

ψ +1

4∆x3

−1 3 −3 1−2 6 −6 2−1 3 −3 1

ψ

+α

4∆x

[−1 0 1−1 0 1

](∣∣∣∣1

4

[1 11 1

]ψ

∣∣∣∣2

1

4

[1 11 1

]ψ

)= 0.

(77)


A "modification" of the Preissman scheme: The last fourequations in MS formulation of CMKDV equation contains notime derivatives, so they can be discretized by omitting thetime average Mt.

The 8-point scheme for CMKDV eq.

DtM3xψ + D3

xMtψ + αDxMx(|MtMxψ|MtMxψ) = 0. (78)

In finite difference stencil format the 8-point scheme (78) isgiven by

1

8∆t

[1 3 3 1−1 −3 −3 −1

]ψ +

1

2∆x3

[−1 3 −3 1−1 3 −3 1

]ψ

+α

2∆x

[−1 0 1

](∣∣∣∣

1

4

[1 11 1

]ψ

∣∣∣∣2

1

4

[1 11 1

]ψ

)= 0.

(79)


The narrow box scheme for CMKDV eq.

1

2∆t

[1 1−1 −1

]ψ +

1

2∆x3

[−1 3 −3 1−1 3 −3 1

]ψ

+α

∆x

[−1 1

](∣∣∣∣

1

2

[11

]ψ

∣∣∣∣2

1

2

[11

]ψ

)= 0.

(80)

Linearized equations and Dispersion relations

Dispersion relations of the 8−point scheme and the Narrowbox scheme


The "good" Boussinesq equation

wtt = −wxxxx + wxx + (w2)xx (81)

Putting w = u − 12 and then vt = −uxxx + (u2)x

(ut

vt

)=

(vx

−uxxx + (u2)x

)=

(0 ∂x

∂x 0

)(δHu

δHv

)(82)

with the Hamiltonian

H(u, v) =1

2

∫ ∞

−∞

(v2 +

2

3u3 + (ux)

2

)dx.


Semi-discrete system of ODEs

Using central finite differences in spatial domain yields

d

dtUi(t) =

Vi+1 − Vi−1

2∆x, (83)

d

dtVi(t) = −

Ui+2 − 2Ui+1 + 2Ui−1 − Ui−2

2(∆x)3+

U2i+1 − U2

i−1

2∆x.(84)

0 0 01 1/2 1/2

A 1/2 1/2

0 1/2 01 1/2 0

B 1/2 1/2

Table: The two-stage Lobatto IIIA-IIIB pair as a partitionedRunge-Kutta method.

Applying Lobatto IIIB to (83) and Lobatto IIIA to (84), andeliminating the external stage vectors in the application yieldsthe


The explicit symplectic scheme for the "good" Boussinesqequation

un+1i − 2un

i + un−1i

∆t2= −

uni−3 − 2un

i−2 − uni−1 + 4un

i − uni+1 − 2un

i+2 + uni+3

4∆x4

+(un

i−2)2 − 2(un

i )2 + (un

i+2)2

8∆x2.


Multisymplectic Integration

Mzt + Kzx = ∇zS(z). (85)

with the state variable z = (u, p, v, q)T and

M =

0 −1 0 01 0 0 00 0 0 00 0 0 0

,K =

0 0 1 00 0 0 1−1 0 0 00 −1 0 0

,

S(z) = −1

3u3 +

1

2(v2 + q2).


Semi-discrete system

Partitioning the state variables z = (u, p, v, q)T into twopartitions z(1) = {u, p} and z(2) = {v, q} and applying theLobatto IIIA-IIIB PRK discretization in space lead to a systemof ODE’s for the Boussinesq equation

d

dtu = Bp,

d

dtp = −Bu + f (u) (86)

where

B =1

∆x2

−2 1 0 · · · 11 −2 1 · · · 0...

. . ....

0 · · · 1 −2 11 · · · 0 1 −2

.


Explicit multisymplectic scheme

In order to obtain an explicit time integrator the partition of thestate variables z(3) = {u, v} and z(4) = {p, q}, then apply thesecond order Lobatto IIIA-IIIB method for the partitionedsystem

un+1i − 2un

i + un−1i

∆t2= −

uni−2 − 4un

i−1 + 6uni − 4un

i+1 + uni+2

∆x4

+(un

i−1)2 − 2(un

i )2 + (un

i+1)2

∆x2. (87)

Linearized equations and Dispersion relations:

Dispersion relations of the symplectic and multisymplecticschemes for the Boussinesq equation


Strongly coupled Nonlinear Schrödinger Equation

iu + βuxx +[α1|u|

2 + (α1 + 2α2)|v|2]

u + γu + Γv = 0,

iv + βvxx +[α1|v|

2 + (α1 + 2α2)|u|2]

u + γv + Γu = 0,

Initial conditions: u(x, 0) = u0(x), v(x, 0) = v0(x)

Boundary Condition: Periodic

β, α1, α2, γ, and Γ are scalar constants

Mass conservation∂∂t

∫∞

−∞

(|u|2 + |v|2

)dx = 0

Energy conservation

1

2

∂

∂t

∫ ∞

−∞

−β(u2

x + v2x

)+ α1

2

(|u|4 + |v|4

)+ (α1 + 2α2)

[|u|2|v|2

]

+γ(|u|2 + |v|2

)+ 2ΓRe{uv}dx = 0


Explicit ODE (B.N.Ryland, R.I.McLachlan 2007)

Applying an r-stage Lobatto IIIA-IIIB PRK discretization inspace to the PDE leads to a set of explicit local ODEs in time inthe stage variables associated with q.

K =

−I 1

2(d1+d2)

I 12(d1+d2)

0d1

, L =

Id1

0d2

−Id1

d1 = n − rank(K), d2 = n − 2d1 ≤ d1, z(1) ∈ Rd1+d2 , z(2) ∈ R

d1 ,

z = (q v p), S(z) = T(p) + V(q) + V(v)

T(p) =1

2ptβp, V(v) =

1

2vtαv, s.t.|β| 6= 0, |α| 6= 0


Multisymplectic Formulation of SCNLS: Kzt + Lzx = ∇zS(z)

u = p + iq, v = µ+ iξ, βux = b + ia, βvx = d + ic

z = (p, µ, q, ξ, b, d, a, c)T

K =

0 −I2 0 0I2 0 0 00 0 0 00 0 0 0

L =

0 0 I2 00 0 0 I2

−I2 0 0 00 −I2 0 0

S(z) = −[

α1

4 (p2 + q2)2 + α1

4 (µ2 + ξ2)2 +(

α1+2α2

2

)(p2 + q2)(µ2 + ξ2)

]

−γ2 (p2 + q2 + µ2 + ξ2) − Γ(pµ+ qξ) − 1

2β (a2 + b2 + c2 + d2)

Explicit ODE for SCNLS, d1 = 4, d2 = 0

Partition z : z(1) = (p, µ, q, ξ)T, z(2) = (b, d, a, c)T

S(z) = T(p) + V(q)

T(p) = 12 pTβp = − 1

2β (a2 + b2 + c2 + d2), β = − 1β I4


Spatial semi-discretization

With this partitioning, SCLNS equation satisfies the requiredconditions so that semi-discretization in space with Lobatto IIIA-IIIBmethod yields explicit an ODE in time.Discretizing the SCNLS equation in space by applying 2-stageLobatto IIIA-IIIB PRK method

Lobatto IIIA :z(1) = {p, µ, q, ξ}T

Lobatto IIIB: z(2) = {b, d, a, c}T

and eliminating the variables in z(2), gives 4 ODEs for each elementof z(1).


∂tqi = β∆x2 (pi−1 − 2pi + pi+1) +

[F(z(1))

]pi + γpi + Γµi

∂tpi = − β∆x2 (qi−1 − 2qi + qi+1) +

[F(z(1))

]qi − γqi − Γξi

∂tξi = β∆x2 (µi−1 − 2µi + µi+1) +

[G(z(1))

]µi + γµi + Γpi

∂tµi = − β∆x2 (ξi−1 − 2ξi + ξi+1) +

[G(z(1))

]ξi − γξi − Γqi

where

F = α1(p2i + q2

i ) + (α1 + 2α2)(µ2i + ξ2

i )

G = α1(µ2i + ξ2

i ) + (α1 + 2α2)(p2i + q2

i )

Note that this amounts to replacing the pxx, qxx, µxx,and ξxx, terms in the SCNLS equation by the centraldifferences.


Semi-discrete MSCL

These ODEs satisfies the semi-discrete multisymplecticconservation law

∂t (dpi ∧ dqi + dµi ∧ dξi)

+ β∆x2 [(dqi+1 + dqi−1) ∧ dqi + (dpi+1 + dpi−1) ∧ dpi

+(dξi+1 + dξi−1) ∧ dξi + (dµi+1 + dµi−1) ∧ dµi] = 0


Time integration

Using the same partitioning z(1) = {p, µ, q, ξ}T andz(2) = {b, d, a, c}T for time discretization yields a RK method inz(1), such as implicit midpoint rule. (J. Hong et al. 2005)(B.Ryland, et al..2007)

Here, we choose the partitioningz(3) = (p, µ, b, d), z(4) = (q, ξ, a, c) to get an explicit integrator.

For systems of ODEs y = f (y, z), z = g(y, z), applying Lobatto IIIA tothe y variable and Lobatto IIIB to the z variable is known as thegeneralized leapfrog method:

yn+1/2 = yn + ∆t2 f (yn+1/2, zn)

zn+1 = zn + ∆t2

[g(yn+1/2, zn) + g(yn+1/2, zn+1)

]

yn+1 = yn+1/2 + ∆t2 f (yn+1/2, zn+1)

We apply second order Lobatto IIIA-IIIB with this partitioning:

Lobatto IIIA : z(3)

Lobatto IIIB : z(4)


Explicit multisymplectic integrator for SCNLS

With the above partitioning of the variables, explicitmultisymplectic integrator for SCNLS can be constructed byapplying 2-stage Lobatto IIIA-IIIB discretization in space andgeneralized leap-frog in time.


Numerical Results: CNLS...Strong coupling Parameter Γ = 0

iu + uxx +[|u|2 + |v|2

]u = 0,

iv + vxx +[|v|2 + |u|2

]u = 0,

Initial conditions :u(0, x) = u0(x), v(0, x) = v0(x)

Boundary Condition: Periodic(-Enables long termcomputations-)

tol = 1.0e − 05 for convergency of Newton method.

The space interval [xL, xR] is discretized by N + 1 uniformgrid points with grid spacing ∆x = h = (xR − xL)/N

t ∈ [0,T]


Single Solitary Wave Solution

iu + uxx +[|u|2 + |v|2

]u = 0,

iv + vxx +[|v|2 + |u|2

]u = 0,

Exact Solution

u(t, x) = sech(x − 2vt)ei(vx−(v2−1)t)

v(t, x) = sech(x − 2vt)ei(vx−(v2−1)t)

Initial condition

u(0, x) = sech(x)ei(vx)

v(0, x) = sech(x)ei(vx)


Rate of convergence

The accuracy is measured by using the L∞ error norm:

‖ER‖∞ = max1≤m≤N {|‖u(tn, xm)‖ − ‖pnm + iqn

m‖|}

rate of convergency ≈ln(ER(h2)/ER(h1)

ln(h2/h1)

where ER(h) is the L∞-error.

−20 ≤ x ≤ 20, ∆t = 0.001, T = 2, v = 1.0

h N L∞ Order

0.5 80 0.7062500 -0.2 200 0.0650605 2.60250.1 400 0.0158917 2.03350.05 800 0.0039189 2.0197


Thank you

joint works with prof.dr.bülent karasözen ayhan aydın...

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