Konrad-Zuse-Zentrum
für Informationstechnik Berlin
Takustraße 7
D-14195 Berlin-Dahlem
Germany
T HORSTEN H OHAGE , F RANK S CHMIDT
On the Numerical Solution of Nonlinear Schrödinger
Type Equations in Fiber Optics
ZIB-Report 02-04 (Januar 2002)
On the Numerical Solution of Nonlinear Schrödinger
Type Equations in Fiber Optics
Thorsten Hohage1 , Frank Schmidt
Abstract
The aim of this paper is to develop fast methods for the solution of nonlinear Schrödinger
type equations in fiber optics. Using the method of lines we have to solve a stiff system of
ordinary differential equations where the eigenvalues of the Jacobian are close to the imaginary axis. This is usually done by a Split Step method. Here we consider the extrapolation
of Split Step methods with adaptive order and step size control. For more complicated nonlinearities, in particular stimulated Raman scattering, Split Step methods are less efficient
since symmetry is either destroyed or requires much additional effort. In this case we use
implicit Runge Kutta formulas of Gauß type. The key point for the efficient implementation of these methods is that the system of nonlinear algebraic equations can be solved
without setting up the Jacobian. The proposed methods are compared to other methods,
in particular exponential integrators, the method of Marcuse, and the method of Blow and
Wood.
1 Introduction
The increasing demand for information transmission has stimulated a large amount of research
in fiber-optic communication systems. A bottle-neck in the simulation of optical networks is
the fast and reliable numerical solution of the pulse-propagation equation in optical fibers. Due
to the use of wavelength division multiplexing (WDM) with more and more channels, nonlinear
effects such as self phase modulation (SPM), cross phase modulation (XPM), four wave mixing,
and the stimulated Raman effect play an increasing role.
The simplest model for pulse propagation in optical fiber including nonlinear effects is the
nonlinear Schrödinger equation:
∂
B
∂z
i
β2 ∂ 2
B iγ B 2 B
2
2 ∂t
(1)
Here B z t is the amplitude modulation function of the rapidly oscillating electric field in a
coordinate system moving with the signal (cf. section 2). For γ 0 eq. (1) is identical to the
free-space linear Schrödinger equation from quantum mechanics, except that the roles of the
time and the space variable are exchanged.
Approximating (1) by the method of lines leads to an ordinary differential equation
B z i
β2
DB z iγ B z 2
2
B z
(2)
where B j z B t j z and where the matrix D approximates
∂2 ∂t 2 . Here t j are given collo
cation points in time, and the absolute value function
is applied componentwise. Since the
spectrum of the differential operator is given by σ 0 5iβ2 ∂2 ∂t 2 it : t 0 , the eigenvalues of iβ2 D will lie on (or close to) the imaginary axis, and their size will grow as the time
1 supported
by DFG grant number DE293/7-1
1
discretization is refined. This implies that (2) is a stiff differential equation which should be
solved by an A-stable method. The second Dahlquist barrier A-stable linear multistep methods
of order 2 cannot exist. Since we have large eigenvalues on the imaginary axis, A α -stable
methods with α 900 are unstable unless very small space propagation steps are used. The
same holds true, of course for explicit Runge-Kutta methods.
In this paper we propose implicit Runge-Kutta methods of Gauß-type and show how the
nonlinear systems of equations can be solved efficiently.
2 Modelling pulse propagation in optical fibers
We start with the derivation of the equation descreibing pulse propagation in optical fibers.
Our presentation follows [1, Chapter 2], but we put a greater emphasis on the Raman effect and
pulses with a large spectral bandwidth. Both points are nicely discussed in [2, 8]. Unfortunately,
there exist different sign conventions. We follow the IEEE convention. The other convention is
1 by i and taking the complex conjugate of all quantities.
obtained by replacing i
The electrical field E t x in optical fibers is governed by the equation
curl curl E
1 ∂2 E
c2 ∂t 2
∂2 P
∂t 2
µ0
(3)
which follows from Maxwell’s equations and the relation D ε0 E P if it is assumed that there
are no free charges and currents and that the magnetic polarization is zero. As usual, ε 0 is the
vacuum permittivity, µ0 is the vacuum permeability, and c ε0 µ0 1 2 is the speed of light.
The electric polarization P is a nonlinear function of E, which we decompose into a linear part
PL and a nonlinear part PNL . For isotropic materials PL is given by
ε0
PL t x t
∞
χ1 t t E t x dt ∞ iωt 1
0 e χ t
1
with a scalar valued function χ1 . We set ε ω x 1 2π
nonlinear polarization is given in good approximation by
ε0 E t x PNL t x 3
t
∞
χ 3 t t E t x
χ g
τ τ
e sin t τ
τ τ
where the kernel χ 3 is of the form χ 3 t gR t 3
3
R
χK δ t 2
1
2
2
t τ2
2
1 2
R t
2
dt. For optical fibers the
dt (4)
with
1 3
The parameters χK and χR determine the strength of the Kerr effect and the stimulated Raman
effect. Typical parameter values in the function gR are τ1 0 0122 ps and τ2 0 032 ps.
We assume that E and PNL are polarized such that E x̂E and PNL x̂PNL and that ∇ E 0.
Then the identity curl curl ∆ grad div yields the Helmholtz approximation
∆Ê ω x ω2 ε ω x Ê ω x c2
2
µ0 ω2 P̂NL ω x (5)
1
iωt dt. Let z be the propagation direction,
for the Fourier transform Ê ω x : 2π
IR E t x e
i.e. let ε ω x y z be independent of z. We make the separation ansatz
Ê ω x y z ℜF x y V̂ ω z (6)
For a single-mode fiber the eigenvalue problem
∆F x y ω2 ε ω x y F x y
c2
iα ω 2
β ω 2
F x y
(7)
has exactly one eigenvalue λ β ω iα ω 2 2 with negative real part for all relevant
frequencies ω. β is called the propagation constant of the guided mode, and α is called attenuation constant or fiber loss. The frequency dependence of α may be neglected in the following.
In the linear case we easily find the separated equation ∂2z V̂ β iα 2 2V̂ and the solution
e iβ ω α 2 z F x y . Hence, the optical power E 2 decays like e αz which exÊ ω x y z plains the factor 12 in (7).
Strictly speaking, it is not possible to obtain a separated equation for V in the nonlinear case
since F actually depends on ω and is complex valued. Fortunately, both the dependence on ω
and the imaginary part of F are very small and may be neglegted in PNL . Hence, multiplying
(5) by F x y and integrating over x and y we obtain
∂2V̂ ω z ∂z2
β ω
iα
2
2
V̂ ω z A
ω2
p̂ ω z c2
(8)
with
A:
and
p t z
V t z
F x y
F x y
t
4 dx dy
2 dx dy
∞
χ3 t t V t z 2
dt Now we assume that V t z is of the form
1
B t z e i ω 0t
2
V t z
where β0
1
B t z e
2
β0 z
i ω0 t β0 z
(9)
β ω0 with a slowly varying envelope function B t z satisfying
B̂ ω z for ω
0
ω0
6
(10)
Inserting (9) into (8) and picking out the terms with nonvanishing Fourier transform in the
interval ω20 3ω2 0 yields
∂
∂z2
2iβ0
∂
∂z
β ω iα
2
2
β20 B̂ ω z 3
ω2
q̂ ω z c2
(11)
where ω
ω0
ω and
3
q t z
The factor
1
4
3χK B t z
8
AB t z 3
χR
4
2
t
∞
gR t t B t z 2
dt comes from neglecting the term
B t z e i ω0 t
β0 z
t
∞
gR t t e2iω0
Bt
t t
z 2 dt which is almost zero due to our assumption (10). We neglegt the interaction between forward
and backward traveling waves and formally factorize (11) to obtain
i∂
∂z
β
0
ω
n2
c
i∂
∂z
q̂ ω ω
n2
c
β0 q̂ ω B̂ ω z 0
c
β ω iα 2 where n ω ω is the (linear) effective refractive index. q̂ can be interpreted as
the square of the nonlinear contribution to the refractive index, and it is usually much smaller
than n2 . Hence, we approximate the square root by its first order Taylor approximation and
neglegt the first differential operator corresponding backward traveling waves to obtain
i∂
B̂ ω z ∂z
β ω
iα
2
β0 B̂ ω z ω0 ω q̂ ω z cn ω0 ω Now the inverse Fourier transform yields the propagation equation for B
∂
B t z
∂z
∂
∂t
∂
τsh i
∂t
i β ω0 i
iγ 1 iα
2
where
2χ 3
γ:
g t
ω0 3χK
3
R A ω0 8cn ω0 β0 B t z t
∞
2
dt B t z
τsh :
g t t B t z
1
ω0
d
A ω
ln
dω n ω 1 fR δ t 3
f R gR t fR
2χ 2χR
3
3
R
3χK
The derivative term in the definition of τsh can usually be neglected. Moreover, it is sufficient
to consider the first few terms of the Taylor expansion of β at ω0 :
β ω0
ω β0
β1 ω β2 2
ω
2
β3 3
ω
6
where β j : β j ω0 . Here β1 can be eliminated from the equation by using a coordinate
system which moves with the signal at the speed of the group velocity 1 β 1 . The function
B̃ t z : B t β1 z z satisfies
∂
B̃ t z ∂z
D
i
∂
∂t
4
N B̃ z B̃ t z (12)
where
D ω i β ω0
:
N B
:
iγ 1 ω β0 β1 ω
∂
τsh i
∂t
t
∞
iα
2
g t t B t 2
α
2
dt β2 2
ω
2
i
β3 3
ω
6
We will always drop the tilde on B below. Typical values of β2 and γ are β2 20 ps2 km and
γ 2 W km 1 . If all other parameters are zero, (12) reduces to (1).
Let us summarize the most important assumptions we have made in the derivation of equation (12) and which restrict its validity in practice:
We assumed that the electric field is polarized in one direction x̂ in the entire fiber.
We ignored backscattering, in particular the Brilloun effect.
We only considered single-mode fibers.
3 Time Discretization and Solution of the Linear Schrödinger
equation
The easiest way to discretize the time variable in the nonlinear Schrödinger equation is to use
the Fast Fourier Transform (FFT). The linear Schrödinger equation (γ 0) can be integrated
exactly in the Fourier domain. However, since the nonlinear part has to be integrated in the
space domain, at least one FFT and one inverse FFT transform have to be applied in each
space propagation step. Since large signals with up to several hundred thousands of unknowns
have to be propagated in practice, the FFT operations dominate the total cost of the integration
process. Therefore, attempts have been made to find alternatives. Some approaches are discussed in subsection 3.2. In this report we mainly focus on space discretization and always use
FFT. However, the methods discussed in the following section can also be combined with other
time discretization schemes. Note that FFT introduces an artificial periodization of the model.
Therefore, the time interval has to be chosen sufficiently large to obtain reliable results.
3.1 Fast Fourier Transform
We solve the partial differential equation (12) by the method of lines. Given an equidistant
C n denote the vector given by
mesh of time points t j tmin j∆t, j 1 n, let B z B z ti . The partial differential equation (12) is approximated by the ordinary differenBi z tial equation
F
B z N B
:
1
D ω F
γ I τsh iF
1
N B B z fR F
2 1 ω F diag 1 fR B 2
(13)
1
F f F B 2 Here F is the Fast Fourier Transform,
ω:
2π n
0 1 n∆t
2
1 5
n
2
T
and denotes the componentwise multiplication of vectors. Moreover D, , and 2 are applied
componentwise.
In some cases it is advantagous to work with the equivalent differential equation for B̂ : F B
given by
D ω
B̂ z F N B F
1
B̂ z (14)
3.2 Other methods
One alternative to FFT is to work in the time domain and approximate the differential operators
∂j
by finite differences or finite elements. In this case one has to cope with the problem of
∂t j
numerical dispersion: High frequencies are propagated with wrong speeds in the numerical
approximation. Therefore, either a very fine discretization or high order elements have to be
used. The use of wavelets is studied in [16].
A different approach has been suggested by Plura [18]. The idea is to approximate the action
of the matrix exp hF 1 diag D ω F by an Infinite Impulse Response (IIR) filter.
4 Analysis of Split Step Methods
4.1 Introduction
We consider an ordinary differential equation of the form
B z LB z where L and N B are matrices.
If N N X is constant, then
B z
N B z B z ehL
h
hN
B z
(15)
(16)
In the simple Split Step method (cf. Table 1) the approximation
ehL
hN
ehL ehN is used. It follows from the Baker-Campbell-Hausdorf formula (cf. e.g. [12]) that the dominant
error contribution in this approximation is due to the term h2 LN NL , and that the simple Split
Step method is locally of order O h2 , i.e. globally of order O h . For constant N it can be seen
that the accuracy of the Split Step method can be increased by one order if the exponential term
are arranged in a symmetric manner. In the nonlinear case this so-called full Split Step method
(cf. Table 1) requires about twice as much memory and computation time as the simple Split
Step method since Bh z has to be computed and memorized in each step. It is not possible to
combine the factors eh 2L of two subsequent steps to one factor ehL . This is the case for the
reduced Split Step method (cf. Table 1) where N Bh z is replaced by N eh 2L Bh z in the
exponent. The reduced Split Step method requires about as much memory and computation
time as the simple Split Step method. However, since the true potential N B h z is replaced
by N X , we may expect that it to be less accurate than the full Split Step method. The plots
in Figure 2 show that this is not the case except for the first order soliton. We will explain this
observation in the next section.
6
An implicit version of the Split Step method has been suggested by Agrawal [19, 1] (cf. Table 1). The implicit equation can be solved by the fixed point iteration
Bh z X0
h
Xn
h
e 2 Le 2
1
e
N Xn
N Bh z
h
2L
Bh z (17)
The following lemma guarantees the convergence of this iteration for sufficiently small step
sizes h.
Lemma 1 Let N : C n C n n be continuously differentiable as a mapping from IR 2n to IR2n and let B C n . Then there exists a constant h0 0 such that the fixed point equation
C n for all h
has a unique solution X
X
h
h
e 2 Le 2
X
N X
N B
h
2L
B
(18)
h0 . The iteration (17) converges to this solution, and
Xn
e
2n ,
θn
1 θ
X1 X0
If L is skew-Hermitian, then h0 and θ do not depend on L.
Proof. We use the Banach
fixed point theorem (cf. [7]). Let f X denote the right hand side
of (18),
and
let
R
:
2
B
.
We
have to show that f is a contractive mapping form B R : X
C n : X R into itself for h h0 . The fact that f maps BR into itself for sufficiently small h
follows from the boundedness of N and exp on compact sets. We have
f X1 f X2 h hL
e2
2
where X t : tX1
C n n is given by
h
N X t N B
2
e
1
D exp
0
hL
2
h
e2
N X1
N B
h
N X1
e2
h
e 2 LB
DN X t N B
h
X1 X2 dt e 2 L B
1 t X2 . The derivative of the matrix exponential function exp : C n D exp X Y
n
1
exp tX Y exp tX dt
0
(cf. [12, page 40]). It follows that there exist constants C and h0 such that D exp h 2 N X t N B C and
f X1 f X2 hC X1 X2
for all X1 X2 BR and 0 h h0 . Hence, f is contractive for h min 1 C h0 with
contraction
factor θ : hC. The last statement follows from the fact that eh 2L is unitary, i.e. eh 2L
1 if
L is skew-Hermitian.
4.2 Adjoint and Symmetric Methods
We consider a general ordinary differential equation
B z f z B z 7
(19)
simple SS
Bh z
full SS
Bh z
reduced SS
Bh z
Agrawal SS
Bh z
B z
e B z
X X e B
e
h
ehL ehN
h
e 2 L ehN
h
e 2 L ehN
h
e 2 Le 2
h
h
h
h
Bh z
h h
2L
Bh z
X
h hL
2
N Bh z h
N Bh z
h z
hL
2
Bh z Table 1: Versions of the Split Step method
with a function f which satisfies a Lipschitz condition with respect to the second variable. (19)
induces an evolution Ψ. For z1 z2 IR, Ψz2 z1 B0 is defined as B z2 where B is the solution to
the initial value problem (19) with the initial condition B z1 B0 . Note that the evolution Ψ
satisfies
Ψz z B B
d
Ψz h z B f z B dh
Ψ z3 z2 Ψ z2 z1 B Ψ z3 z1 B
(20)
and that these properties characterize the evolution uniquely (cf. [6]). For a numerical solution of the differential equation (19) the continuous evolution Ψ is approximated by a discrete
evolution Φ. E.g., the discrete evolution of the simple Split Step method is
ΦsimSS
z hzB:
ehL ehN
B
B
A discrete evolution which satisfies the first two conditions in (20) is called consistent. In
general, a discrete evolution does not satisfy the third condition (otherwise it would be the exact
evolution!). However, in some cases a discrete evolution may satisfies the third condition for
the special case z3 z1 .
Definition 2 Let Φ be the discrete evolution of some method. The adjoint evolution Φ is
defined by
Φz
h z Φz z h B
B
(21)
Φ is called symmetric if Φ
Φ. The method described by Φ is called adjoint method. A
method is called symmetric if its discrete evolution is symmetric.
The following theorem on symmetric methods is proved in [9, Theorem 8.10].
Theorem 3 Suppose that f and Φ are sufficiently smooth, and that Φ is consistent and symmetric. Then the global error satisfies
Bh z B z for M
IN as h
e 2 z h2
e4 z h 2
e2M
2 z
h2M
2
O h2M 0. Here e j are smooth functions.
In particular, a consistent, symmetric method is at least of global order 2.
8
(22)
Φ
B z h
e e
B z
is defined implicitly. Even if N B z h
As a first example, let us compute the adjoint evolution ΦsimSS of the simple Split Step
method. Let Bh z ΦsimSS
e hL e hN Bh z Bh z h . The definition (21) implies
z z h Bh z h simSS
z hz
hN Bh z h
h h
hL
Bh z h
Note that ΦsimSS
N Bh z (e.g. for linear
Schröding equations with a constant potential), the simple Split Step method is not symmetric in general, since the matrices L and N Bh z do not commute. An analogous computation
shows that
ΦfulSS
z h z Bh z Bh z
h
e 2 L ehN
h
The full Split Step method is symmetric if N Bh z
method we obtain
redSS
hz
Φz
h
e 2 LX Bh z e
Bh z h
h hL
2
Bh z N Bh z . For the reduced Split Step
ehN
X
e
X
h
2L
Bh z In general, the reduced Split Step method if not symmetric. However, if N X as in (2), then the diagonal entries of exp hN X have norm 1, and hence
Introducing Y
e
X we find N Y
hN x
ΦredSS
z h z Bh z X
N e
X N X . Therefore,
e Y
Y e B z
N ehN
N X
hN X
h
e2L
X
hN Y
iγdiag X
2
(23)
hL
2
h i.e. the reduced Split Step method is symmetric if N satisfies (23). Finally, it is easy to see that
Agrawal’s method is symmetric.
Let us summarize our results.
Proposition 4
1. The simple Split Step method is not symmetric in general.
2. The full Split Step method is symmetric if N B does not depend on B.
3. The reduced Split Step method is symmetric if N satisfies (23).
4. The Split Step method of Agrawal is symmetric.
4.3 Numerical Results
We compared different versions of the Split Step method for the nonlinear Schrödinger equation
(1) with γ 2 W km 1 and β2 20 ps2 km. A class of test examples with an explicitely
known solutions are solitons of order ν IN. The initial data are given by
B 0 t ν Psech
γP
t
β2
We tested the Split Step methods for the first and second order soliton with P
propagation distance of 50 km.
9
(24)
0 1W and a
amplitude and real part after 0 km
0.2
intensity
0.1
0
−0.1
−0.2
−1000
−800
−600
−400
−200
0
time [ps]
200
400
600
800
0
0.2
frequency [THz]
0.4
0.6
0.8
frequency
0
intensity
10
−5
10
−0.8
−0.6
−0.4
−0.2
1
Figure 1: WDM test signal with 2 channels
1st order soliton
5
2nd order soliton
5
10
10
number of steps
number of steps
4
10
3
10
simple Split Step
full Split Step
reduced Split Step
Agrawal
2
10
1
10
0
10
2
4
10
6
10
3
10
simple Split Step
full Split Step
reduced Split Step
Agrawal
2
10
−2
10
8
10
4
10
10
precision=1/(rel. L2−error)
0
10
2
4
10
10
6
10
precision=1/(rel. L2−error)
1 channel
WDM 2 channels
4
10
4
3
10
simple Split Step
full Split Step
reduced Split Step
Agrawal
2
10
number of steps
number of steps
10
3
10
2
10
simple Split Step
full Split Step
reduced Split Step
Agrawal
1
0
10
2
10
10
0
10
4
10
precision=1/(rel. L2−error)
2
10
4
10
6
10
precision=1/(rel. L2−error)
Figure 2: Comparison of different versions of the Split Step method
10
8
10
Moreover, we consider a wavelength division multiplexing (WDM) system with 2 channels
(cf. Figure 1). The first channel, centered at 0 3 THz, carries a signal encoding the bit sequence
110011101. Initially the signal is located in the left part of the time interval, but due to dispersion it moves to the right in the given coordinated system. The signal in the second channel,
centered at 0 3 THz, corresponds to the bit sequence 11100101 and moves to the left. After a
distance of 18 km the first signal has passed the second signal.
In Figure 2 we plotted the number of steps K Z h over the precision p 1 B Z Bh Z with a logarithmic
scale on both axes. According to (22) we have asymptotically
1
1
2
K , i.e. that the slope is asymptotically 1 2. For first order acculn p 2 ln Z
e2 Z rate methods the slope is asymptotically 1. In all these test examples the reduced Split Step
method and the Agrawal Split Step method are second order accurate. The full Split Step
method is only second
order
accurate for the special case of the first order soliton since here
2 is independent of z. The simple Split Step method is only first order
N B z t iγ B z t accurate asymptotically. However, for large step sizes it yields almost the same results as the
reduce Split Step methods. The explanation is that the simple Split Step method yields the
same result as the reduced Split Step method applied to the initial data e h 2L B 0 t if the result
if multiplied by e h 2L . However, this is only true for constant step sizes.
4.4 Conclusion
We have discussed criteria for symmetry of Split Step methods and consequences. Symmetry
guarantees second order accuracy and is advantageous for exptrapolation methods discussed in
the next section. If only the Kerr effect is considered, the reduced Split Step method yields the
best results. For more complicated nonlinearities Agrawal’s version of the Split Step method
should be used.
5 Extrapolation of Split Step Methods
5.1 Introduction
Extrapolation of a “basic” method consists in first propagating the signal over a certain distance
several times with different step size and then interpolating the results.
Theoretically, extrapolation methods are based on an asymptotic expansion of the following
form
Bh z B z
e κ z hκ
e2κ z h2κ
e h κM 1
κM 1
O hκM (25)
It can be shown that such a relation is always satisfied with κ 1 under certain smoothness
assumptions (cf. [9, Theorem 8.1]). Of course, the first error terms in (25) vanish for higher
order methods. Extrapolation is particularly efficient if κ 1. We have seen in Theorem 3 that
(25) is satisfied with κ 2 for symmetric methods.
We introduce a vector pz of polynomials of degree
M 1 by
pz hκ : B z eκ z hκ
e2κ z h2κ
e h κM 1
κM 1
Note that pz 0 B z is the exact solution. In order to approximate pz 0 by interpolation, we
evaluate the polynomial pz at M distinct points h n j κ with an error of order O hκM . Here
11
T1 1
T2 1
..
.
TM
T2 2
..
.
T
TM
11
TM 1
1M 1
TM M
MM 1
Figure 3: Illustration of the Aitken-Neville Algorithm.
S
n1 n2 nM is a given step sequence. The interpolation can be done efficiently by the
Aitken-Neville algorithm (cf. [7]).
Given some basic method with discrete evolution Φ satisfying (25) and a step sequence S
n1 n2 nM , we define an extrapolated evolution ΦS of global order κM by the following
algorithm:
Extrapolation of a discrete evolution Φ with step sequence S .
1. Compute T j 1 : Φz
ph h n j κ
O
n1 1
n1 h
hz
κM
h !)
Φz
2. (Aitken-Neville Algorithm) For k
2
nj h
Tj k
1 1
Φz
1
nj h
M 1 and j
: Tj k
1
nj h
z B for
z
Tj k Tj
nj κ
nj k k
1k
j
1 M. (Note that T j 1
1 M compute
(26)
1
3. Set ΦSz
h zB :
TM M .
The Aitken-Neville Algorithm has the advantage that it provides a complete table of numerical results T j k which can be used as error estimates for step size and order selection. It is easy
to show that T j k represents a method of order κk.
Several step sequences S have been suggested:
Romberg sequence: 1 2 4 8 16 32 64 Burlisch sequence: 1 2 3 4 6 8 12 16 24 32 harmonic sequence: 1 2 3 4 5 6 7 We have found the harmonic sequence to be most efficient for the nonlinear Schrödinger equation.
5.2 Step Size Selection
Let us first consider extrapolation of fixed order with variable step size. Since a control of the
global error is hard to realize, we try to control the local error TM M B Φz h z B . Here we use
the norm
B :
1
n
n ∑
j 1
12
Bj
2
0.12
step size [km]
0.1
0.08
0.06
0.04
0.02
0
0
2
4
6
8
10
position z [km]
12
14
16
18
5
realized
suggested
4.5
order
4
3.5
3
2.5
2
0
2
4
6
8
10
position z [km]
12
14
16
18
Figure 4: Order and step size selection for the extrapolation of Split Step methods
An estimate of TM M B Φz h z B would require a better approximation to Φz h z B than TM M .
However, if we had such an approximation we should use it for the further computations. Therefore, we try to control the error of the second best approximation TM M 1 in the Aitken-Neville
scheme. More precisely, we require that
TM
TM M
1M Tol
(27)
where ε is some given tolerence prescribed by the user. The real error TM M B Φz h z B is
usually much smaller. If (27) is violated, we repeat the integration step with a smaller step size.
opt
opt
Otherwise, we compute a new “optimal” step size hM for the following step. hM is chosen
such that TM 1 M TM M Tol . Since
TM
TM M
1M c z hκ M
1
1
we obtain
opt
hM
ρh
TM
Tol
1M TM M
1
κ M 1 1
(28)
The safety factor ρ is included to increase the probability that (27) is satisfied in the next step.
We use ρ 0 8.
5.3 Order Selection
We want to choose the order of the extrapolation method adaptively such that the amount of
work per unit step is minimized under the side condition that a given accuracy is guaranteed. The amount of work of one step of an extrapolation method with step sequence S
13
n1 n2 nM is given approximately by
M
∑ sm AM :
m 1
and the work per unit step is measured by
AM
HM
WM :
Moreover, we introduce the error estimates errM :
TM M 1 TM M Tol which must be 1
to satisfy (27).
We proceed with a detailed description of our combined step size and order selection algorithm (cf. [9, 3, 4]). Assume for some integration step the step size H and the order M 3 have
been suggested. Then we compute the first M 1 lines of the extrapolation scheme in Fig. 3.
1) Convergence in line M 1. If errM 1 1, we accept TM 1 M 1 as numerical solution
and proceed with the new proposed quantities
M 1
M
Mnew
if M
else
3 and WM
0 9 WM
1
2
(29a)
HMnew
HM 1 AM AM
Hnew
1
if Mnew
if Mnew
M 1
M
The expression HM 1 AM AM 1 is an approximation to WM based on the assumption WM WM 1 which can be evaluation without computing line M in the extrapolation tableau.
2) First convergence monitor. If errM 1 1 we use the heuristic estimate errM n1 nM 2 errM
(cf. [9]) to decide if we can expect convergence in line M or M 1. The condition err M 1 1
leads to the criterion
errM
nM
1
1 nM
2
n1
2
(30)
If (30) is satisfied, we reject the step and restart with Mnew and Hnew given in (29). Otherwise
we compute the Mth line of the extrapolation tableau.
3) Convergence in line M. If errM 1 we accept TM M as numerical solution and continue
the integration with the new quantities
Mnew
M 1
M 1
M
if M
if M
else
3 and WM 1 0 9 WM
Mmax and WM 0 9 WM
1
(31a)
Hnew
HMnew
HM AM
1
AM if Mnew
if Mnew
M
M
1
Here Mmax is the largest admissible order of the extrapolation scheme. We used M m ax
this was never realized in our numerical experiments.
14
7, but
1
2nd order soliton
5
nr of FFT operations
nr of FFT operations
4
10
3
10
red. Split Step
Blow and Wood
adapt Extrap 12
adapt Extrap 123
Extrap var. order
2
10
0
10
WDM 2 channels
5
10
10
2
4
10
6
10
8
10
3
10
red. Split Step
adapt Extrap 12
adapt Extrap 123
Extrap var. order
2
10
2
10
10
10
4
10
10
3
4
10
10
5
6
10
10
7
10
8
10
9
10
precision=1/(rel. L2−error)
precision=1/(rel. L2−error)
Figure 5: Convergence of extrapolation methods
4) Second convergence monitor. If errM 1 we again check whether or not we may expect
convergence in line M 1 of the extrapolation tableau. Since now err M is at our disposal, (30)
can be replaced by the more reliable test
errM
nM 1
n1
2
(32)
If (32) is satisfied, the step is rejected and we restart with Mnew and Hnew given by (31). Otherwise we compute line M 1 of the extrapolation tableau.
5) Convergence in line M 1. If errM 1 1 we accept TM 1 M 1 as numerical solution and
continue the integration with
Mnew
Hnew
M 1
M 1
M
if M
if M
else
3 and WM 1 0 9 WM
Mmax and WM 1 0 9 WM
HMnew (33a)
Otherwise we reject the step and restart with Mnew M 1 if M 3 and WM 1 0 9 WM ,
Mnew M else, and Hnew HMnew .
If M 2 is the optimal choice of order, the order tends to flip back and forth between 2 and
3, or it remains at M 3. This is because in (29b) a too large step size appropriate for M 3 is
suggested. Therefore we estimate H3 and W3 by H̃3 ρH n1 n3 2 err2 1 5 and W̃3 A3 H̃3
and replace (29b) by
Hnew
in the important case M
H2
H̃3
if W2
else
0 9 W̃3
3.
5.4 Numerical Experiments
Figure 5 shows the results of the order selection algorithm applied to the 2 channel WDM
problem described in subsection 4.3 (cf. Figure 1). During the overlap of the two signals the
15
order is increased. The convergence of the extrapolation of the reduced Split Step method
with adaptive step size selection and fixed step sequence S 1 2 and S 1 2 3 and the
convergence of the extrapolation method with adaptive step size and order selection are plotted
in Figure 5.
6 Collocation Methods
6.1 Introduction
Let us consider a general differential equation
B z f z B z (34)
We make the ansatz
B̃ z
s
∑ γ jτ j
τh (35)
j 0
with γj C n and require that B̃ satisfies the differential equation at s collocation points 0 c1
c2
cs 1:
B̃ z
ci h fz
ci h B̃ z
ci h i
1 s
(36a)
Moreover, we require
B̃ z B0
h z B0
B̃ z
and set
Φz
(36b)
h
(36c)
To show that this method is a Runge-Kutta method, we introduce the Lagrange polynomials L j
with respect to c1 cs . L j is the (unique) polynomial of degree s 1 satisfying L j ci δi j
for j 1 s. Since B̃ is a vector of polynomials of degree s 1, we have
B̃ z
s
∑ k j L j τ
τh (37)
j 1
with k j
B̃ z
c j h . It follows that
B̃ z
τh B̃ z h
τ
B̃ z
0
th dt
(38)
Inserting (38) and (36b) in (36a) and (36c) yields
ki
fz
ci h B0
s
h ∑ k j ai
j i
1 s
(39a)
j 1
Φz
h z B0
B0
s
h ∑ k jb j
j 1
16
(39b)
with
ai j :
ci
0
L j t dt
and
bj
1
0
L j t dt Note that a collocation method is uniquely determined by the choice of the points c j . The
highest possible global order 2s is achieved if the c j are chosen as Gauß quadrature points.
The corresponding method, which are called Gauß methods, are known to be A-stable and
symmetric (cf. [10]). In our numerical computations we used the Gauß method of order 6.
Other commonly used methods are Radau methods (cs 1: order 2s 1) and Lobatto methods
(c1 0 and cs 1: order 2s 2). The numerical values of the Runge-Kutta coefficients a i j , b j
and c j of the first Gauß, Radau and Lobatto methods can be found in [10] and [6].
6.2 Numerical solution of the nonlinear equations
For a numerical solution of the system of equations (39) we introduce the new unknowns
s
h ∑ ai j k j
yi
j 1
and reformulate (39) equivalently as
s
yi
h ∑ ai j f z
j 1
c j h B0
yj
i
1 s
(40a)
Φz
h z B0
B0
s
∑ d jy j
(40b)
j 1
where d T bT A 1 (cf. [6]). Combining y1 system of equations (40a) as
ys to one large vector Y , we may rewrite the
hF z Y Y
(41)
For the nonlinear Schrödinger equation (14), eq. (40a) has the form
Y
hLY
hN Y (42)
Here L is a block diagonal matrix whose diagonal blocks are D ω j A. As opposed to the fixed
point equation in Agrawal’s method (cf. Lemma 1), eq. (42) should not be solved by a fixed
point iteration. This is because the imaginary parts of the eigenvalues of L get larger and larger
as the time discretization gets finer, and hence the step size h has to be chosen very small to
1 hN Y and use the fixed
ensure convergence. Therefore, we rewrite (42) as Y
I hL point iteration
Yn
1
I hL 1
hN Yn (43)
The iteration (43) can be interpreted as an inexact Newton iteration for (42) where the derivative
matrix is approximated by I hL.
17
2nd order soliton
0.5
step size [km]
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
position z [km]
35
40
45
50
0
5
10
15
20
25
30
position z [km]
35
40
45
50
nr of Newton its
7
6
5
4
3
Figure 6: Step size selection and termination of the Newton iteration for collocation methods
6.3 Stopping criterion and starting values for the Newton iteration
We first derive a stopping criterion for the inexact Newton iteration 43 (cf. [5]). If h is sufficiently small, the updates ∆Yn : Yn Yn 1 , n 1 satisfy
with θ
∆Yn
θ ∆Yn
1
1. Applying the triangle inequality to
Yn
1
Y
Yn
Yn
1
2
Yn
Yn
2
3
yields the estimate
Yn
Y
1
Θ
∆Yn
1 Θ
To obtain a computable error estimate we approximate the convergence rate Θ by the quantities
Θn :
∆Yn
∆Yn
We have thus derived the stopping criterion
Θn ∆Yn
1 Θn
1 n
κTol
2
(44)
for the iteration (43). Since the error in the Newton iteration should be smaller the the discretization error, which is usually close to Tol , the parameter κ should be 1. If (44) is not
satisfied after nmax iterations or if θn 1 for some n, the step is repeated with step size h 2. We
used the parameter values κ 10 4 and nmax 10.
The starting values are computed by extrapolating the polynomials (35) of the previous step.
This saves about 1-2 Newton iterations compared to starting the iteration with the zero vector.
18
4
6
10
solution
initial
4
2
0
−1.5
−1
−0.5
0
0.5
1
1.5
time [ps]
20
10
intensity [W]
1st order soliton with Raman effect
6
x 10
number of steps
intensity [W]
8
0
10
solution
initial
5
10
4
10
Agrawal
Marcuse 4
gauss6
−20
10
3
−40
10
−300
−200
−100
0
100
200
10
0
10
300
frequency [THz]
2
4
10
6
10
10
8
10
precision=1/(rel. L2−error)
Figure 7: Decay of a first order soliton under the influence of the Raman effect
6.4 Step Size Selection
To implement an adaptive step size selecting algorithm we need an estimate of the error. In
analogy to the procedure described in [10, p. 133] for the 3-stage Radau IIA method, we discuss
the construction of an embedded lower order Runge-Kutta method of the form
Φ̂z
h z B0
B0
h b̂0 f z B0 ∑ b̂i f z s
i 1
ci h B0
yi b̂s
1f
z
h Φz
h z B0 with b̂0 0. Given a choice of b̂0 , e.g. b̂0 s 1 1 , we determine the other coefficients b̂i
such that the embedded evolution Φ̂ has the maximal possible order s 2. This is the case if b̂i
j
j
1
satisfy the system of equations ∑si 11 b̂i ci
0 s with c0 : 0 and cs 1 : 1.
j 1 b̂0 c0 , j
The difference err : Φ̂z h z B0 Φz h z B0 serves as an error estimate. It satisfies
hb̂0 f z B0 errsss
hb̂s
1f
z
h Φz
h z B0 s
∑ ei yi
i 1
with e1 es b̂1 b1 gestion for the new step size:
b̂s bs A 1 . We use the following formula to compute a sug-
hopt
0 9
nmax
nmax
1
n
Tol
err
1 s 2
6.5 Numerical examples
The performance of our step control algorithm and the stopping criterion for the Newton iteration is illustrated in Figure 6 for the second order soliton.
The convergence plots in Figure 8 show that the collocation method performs well for signals with a small bandwidth, but not as good for signals with a large bandwidth such as WDM
signals. The reason is that small step sizes are required to approximate the rapid oscillations of
high frequencies suffiently well by polynomials.
We now consider the full propagation equation (12) including the Raman effect. In this
case the propagation of the nonlinearity in Split Step methods can no longer be performed
19
analytically. In principle we could resort to Runge-Kutta methods, but due to rapid oscillations
this is not an efficient options. Agrawal’s implicit version of the Split Step method discussed in
section 4 works significantly better.
As a test example we consider the decay of a first order soliton. Notice the huge red shift
in the spectrum (cf. Figure 7). Due to Raman scattering energy is transfered from higher frequencies to lower frequencies. The convergence plot in figure 7 shows that collocation methods
clearly outperforms Agrawal’s method for this example.
6.6 Conclusion
The efficient implementation of collocation methods of Gauß type for nonlinear Schrödingertype equations including adaptive step size selection and termination of the Newton iteration.
These methods can handle complicated nonlinearities effectively. They are not efficient for
signals with a large spectral bandwidth.
7 Other Methods
7.1 Exponential Integrators
Hochbruck and Lubich have studied numerical integrators which are based on the evaluation of
the exponential of the Jacobian (cf. [13, 14, 15]). These methods are exact for linear differential equations with constant coefficients if the exponential of the coefficient matrix is evaluated
exactly. The idea is to use Krylov subspace approximations to the action of the matrix exponential operator. Although this idea had been considered earlier, Hochbruck and Lubich were the
first to prove that Krylov approximations to exp hA converge much faster than those for the
solution of linear systems I hA x v ([13]).
Moreover, they developed the first exponential integrator of order 4 ([14]). Figure 8 shows
a comparision of their code exp4 and other methods for the nonlinear Schrödinger equation.
The poor performance of exp4 can be explained as follows: Let A be a skew-Hermitian matrix
with eigenvalues in an interval on the imaginary axis of length 4ρ. Then the error in the Arnoldi
approximation of exp hA is not decay essentially for m hρ where m is the dimension of the
Krylov subspace (cf. [13, Theorem 4 and Figure 3.2]). Since ρ is very large in our application,
we either need small step sizes h or a large number m of iterations. In both cases the method is
inefficient.
Hochbruck and Lubich also suggested a Gautschi-type method for differential equations of
the form
B z AB z g B z B 0
B0 B 0 B0 which is second order accurate ([15]). It is possible to adopt this scheme to the nonlinear
Schrödinger equation, but the resulting method is only first order accurate (cf. [11]). We do not
know how to construct higher order methods.
7.2 The method of Marcuse
Eq. (14) can be solved explicitly if N
B̂ z 0. The solution is given by
eD ω z B̂ 0 20
j
1
n
2nd order soliton
5
10
number of steps
4
10
3
10
exp4
Gautschi
Marcuse
red. SS
Extrapolation
gauss 6
2
10
1
10
−2
10
0
10
2
10
4
10
6
10
8
10
10
10
precision=1/(rel. L2−error)
1 channel
5
number of steps
10
4
10
exp4
Gautschi
Marcuse
red. SS
Extrapolation
gauss 6
3
10
2
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
precision=1/(rel. L2−error)
2 channel WDM, DCF
5
number of steps
10
4
10
exp4
Gautschi
Marcuse
red. SS
Extrapolation
gauss 6
3
10
2
10
−2
10
0
10
2
10
4
10
6
10
8
10
10
10
precision=1/(rel. L2−error)
Figure 8: Comparison various methods for the solution of the nonlinear Schrödinger equation
21
2nd order soliton
5
WDM 2 channels, DCF
5
10
10
Extrapolation
Blow and Wood
nr of FFT operations
nr of FFT operations
Extrapolation
Blow and Wood
4
10
3
10
4
10
3
10
2
2
10
−2
10
0
10
2
4
10
6
10
8
10
10
1
10
10
10
10
2
3
10
10
4
5
10
10
6
10
precision=1/(rel. L2−error)
precision=1/(rel. L2−error)
Figure 9: Comparison of the method of Blow and Wood and second order extrapolation
The high frequency components of this solution oscillate rapidly in space. This is undesirable
for a numerical approximation of these components. Now the idea of Marcuse et.al. [17] is to
compute the function
C z : e
Since C z F
Dωz
B̂ z , C satisfies the differential equation
N Bz Bz
Bz
F e C z B̂ z D ω B̂ z e
C z Dωz
iD ω z
1
(45)
This differential equation can be solved by explicit Runge-Kutta methods.
The method of Marcuse yields good results for small step sizes. However, for large step
sizes it is often unstable.
7.3 The method of Blow and Wood
We have seen in subsection 4.2 that the leading local error term of symmetric Split Step methods
is given by e z t h3. It may be assumed that e z t varies slowly with z. Therefore, taking 4
forward steps of size h followed by 1 backward step of size 2h and 4 forward steps of size h
eliminates the leading error term as 4h3 2h 3 4h3 0. This is the method of Blow and
Wood [2]:
ΦBW
2h
Φh Φh Φ
4 times
2h Φh Φ h
(46)
4 times
In total we have to take 9 steps to propagate the signal over the distance distance 6h. The same
amount of steps is needed for the second order extrapolation scheme described in section 5.
Since both methods eliminate the leading error term, it is not surprising that their performance
is almost the same (cf. Figure 9). However, the extrapolation method has the advantage that it
also yields an error estimate, which can be used for adaptive step size selection. Moreover, the
construction of higher order schemes is straightforward. Therefore, we prefer the extrapolation
method from section 5 to the method of Blow and Wood.
22
7.4 Numerical results and conclusions
We have chosen three test examples to compare the different methods (cf. Figure 8): The second
order soliton, a one channel system with P 0 1 W over a propagation distance of 50 km, and
a 2 channel system with damping constant α 0 05 including a dispersion compensating fiber
(DCF).
The results show that exponential integrators are not efficient for the integration of the nonlinear Schrödinger equation since the application of the exponential of the jacobian is too expensive. The method of Marcuse is flexible and often yields good results, but for large step sizes
it tends to be unstable. The method of Blow and Wood is similar to second order extrapolation,
but it is not clear how to contruct higher order methods and adaptive step size selection in an
efficient manner.
In summary, there is no method which performs well in all situations. For high accuracies
and the simple nonlinear Schrödinger equation (1) extrapolation of Split Step methods is most
efficient. For low accuracies it is better to use the reduced Split Step method without extrapolation. If the Raman effect is considered and high powers are involved, collocation methods of
Gauß type yield the best results.
References
[1] G. Agrawal. Nonlinear Fiber Optics. Academic Press, San Diego, 1989.
[2] K. J. Blow and D. Wood. Theoretical description of transient stimulated Raman scattering
in optical fibers. IEEE J Quantum Electronics, 25(12):2665–2673, 1989.
[3] P. Deuflhard. Order and stepsize control in extrapolation methods. Numer. Math., 41:399–
422, 1983.
[4] P. Deuflhard. Recent progress in extrapolation methods for ordinary differential equations.
SIAM review, 27(4):505–535, 1985.
[5] P. Deuflhard. Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive
Algorithms. ., In preparation.
[6] P. Deuflhard and F. Bornemann. Numerische Mathematik II. Walter de Gruyter, Berlin,
New York, 1994.
[7] P. Deuflhard and A. Hohmann. Numerische Mathematik I. Walter de Gruyter, Berlin, New
York, 2nd edition, 1993.
[8] J.-P. Elbers. Modellierung und Simulation faseroptischer Übertragungssysteme. Master’s
thesis, Uni. Dortmund, 1996. Diplomarbeit.
[9] E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff
Problems, volume 8 of Springer series in Computational Mathematics. Springer-Verlag,
Berlin, New York, 2nd edition, 1992.
[10] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems, volume 14 of Springer series in Computational Mathematics.
Springer-Verlag, Berlin, New York, 1991.
23
[11] J. Häring. Entwurf schneller Algorithmen zur Lösung der nichtlinearen SchrödingerGleichung. Master’s thesis, Uni. Dortmund, 1998. Diplomarbeit.
[12] J. Hilgert and K.-H. Neeb. Lie-Gruppen and Lie-Algebren. Vieweg Verlag, Braunschweig,
1991.
[13] M. Hochbruck and C. Lubich. On Krylov subspace approximation of the matrix exponential operator. SIAM J. Numer. Anal., 34:1911–1925, 1997.
[14] M. Hochbruck and C. Lubich. Exponential integrators for large systems of differential
equations. SIAM J. Sci. Comput., 19:1552–1574, 1998.
[15] M. Hochbruck and C. Lubich. A Gautschi-type method for oscillatory second-order differential equations. Numer. Math., 83:403–426, 1999.
[16] T. Kremp, A. Killi, A. Rieder, and W. Freude. Split-step wavelet collocation method
for nonlinear pulse propagation. Technical Report 01/14, Institut für Wissenschaftliches
Rechnen und Mathematische Modellbildung, Univ. Karlsruhe, Universität Karlsruhe,
76128 Karlsruhe, Germany, 2001.
[17] D. Marcuse, A. R. Chraplyvy, et al. Effect of fiber nonlinearity on long distance transmission. J. Lightwave Tech., 9(1):121–128, 1991.
[18] M. Plura. Makromodellierung und schnelle Simulationsmethoden zur Lösung der nichtlinearen Schrödinger-Gleichung. Technical report, Lehrstuhl für Hochfrequenztechnik,
Prof. Voges, Uni. Dortmund, 2001.
[19] M. J. Potasek, G. P. Agrawal, and S. C. Pinault. Analytic and numerical study of pulse
broadening in nonlinear dispersive optical fibers. J.Opt.Soc.Am B, 3(2), 1986.
24