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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
GEOMETRIC INTEGRATION
Simple Harmonic oscillator
md2x
dt2+ kx = 0
k = m = 1 :dx
dt= p,
dp
dt= −x
(1)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
−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 )
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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, . . . )
...
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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).
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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?
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
"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 "
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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).
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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).
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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))
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
Dispersion relations of the Preissman scheme and the sixpoint scheme for the CNLS equation are discussed
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
International Conference on Numerical Analysis and AppliedMathematics(ICNAAM2008),
Kos-Greek(2008)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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).
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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).
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
∂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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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.
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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]
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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)
IN HONOUR OF BÜLENT KARASÖZEN’S 67TH BIRTHDAY
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