text: 17 0.7 Fatunla 0.6 Arnoldi a.u.)0.5 n ( utio0.4 b stri0.3 di y g0.2 r e n E 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Energy (a.u.) FIG. 10. (Color online) Energy distribution resulting from the interaction of the hydrogen atom with a cosine square pulse. Fatunla’s and Arnoldi’s propagators are used. The pulse has a peak intensity I=1014 Watt/cm2, a frequency ω = 0.7 a.u. and a duration of 10 optical cycles. The basis-set of functions used is a set of 100 Coulomb Sturmian functions per angular momentum. Ten angular momenta are included and the non-linear parameter κ of the Coulomb Sturmian functions is equal to 0.3. The Arnoldi propagator uses 5 Krylov vectors and a time step of δt = 0.05 a.u.The relative diﬀerence between both curves − is of the order of 10 3. 0.7 N =4, = 0.3 krylov ! 0.6 N =5, = 0.3 krylov ! a.u.)0.5 n ( utio0.4 b stri0.3 di y g0.2 r e n E 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Energy (a.u.) FIG. 11. (Color online) Energy distribution resulting from the interaction of the hydrogen atom with a cosine square pulse. Arnoldi’s propagator is used. The pulse parameters are as in Fig.10, using 100 Coulomb Sturmian functions per angular momentum. Ten angular momenta are included and the non-linear parameter κ of the Coulomb Sturmian functions is equal to 0.3. For a time step of δt = 0.05 a.u., we compare results when 5 and 4 Krylov vectors are used. distribution require a minimum of 5 Krylov vectors. These calculations performed in a Coulomb Sturmian basis can be further tested by varying their non-linear pa- rameter κ. If instead of using κ = 0.3, we use κ = 0.4, all the other parameters remaining the same, we again obtain a completely ﬂat energy distribution. By increasing the value of the non-linear parameter κ, the value of the eigenenergies increases thereby increasing the stiﬀ character of the problem. To successfully reproduce an accurate energy distribution we now would need to increase the number of Krylov vectors. If on the other hand, we keep the value of κ equal to 0.3 and increase the number of basis functions, converged results are only obtained when 8 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fatunla

text: Arnoldi

text: n

text: b

text: di

text: y

text: r

text: e

text: n

text: E

text: Energy

text: (a.u.)

text: FIG.

text: (Color

text: online)

text: Energy

text: distribution

text: resulting

text: from

text: the

text: interaction

text: of

text: the

text: hydrogen

text: atom

text: with

text: a

text: cosine

text: square

text: pulse.

text: Fatunla’s

text: and

text: Arnoldi’s

text: propagators

text: are

text: used.

text: The

text: pulse

text: has

text: a

text: peak

text: intensity

text: a

text: frequency

text: ω

text: a.u.

text: and

text: a

text: duration

text: of

text: optical

text: cycles.

text: The

text: basis-set

text: of

text: functions

text: used

text: is

text: a

text: set

text: of

text: Coulomb

text: Sturmian

text: functions

text: per

text: angular

text: momentum.

text: Ten

text: angular

text: momenta

text: are

text: included

text: and

text: the

text: non-linear

text: parameter

text: κ

text: of

text: the

text: Coulomb

text: Sturmian

text: functions

text: is

text: equal

text: to

text: The

text: Arnoldi

text: propagator

text: uses

text: Krylov

text: vectors

text: and

text: a

text: time

text: step

text: of

text: δt

text: a.u.The

text: relative

text: diﬀerence

text: between

text: both

text: curves

text: is

text: of

text: the

text: order

text: of

text: N

text: krylov

text: N

text: krylov

text: n

text: b

text: di

text: y

text: r

text: e

text: n

text: E

text: Energy

text: (a.u.)

text: FIG.

text: (Color

text: online)

text: Energy

text: distribution

text: resulting

text: from

text: the

text: interaction

text: of

text: the

text: hydrogen

text: atom

text: with

text: a

text: cosine

text: square

text: pulse.

text: Arnoldi’s

text: propagator

text: is

text: used.

text: The

text: pulse

text: parameters

text: are

text: as

text: in

text: using

text: Coulomb

text: Sturmian

text: functions

text: per

text: angular

text: momentum.

text: Ten

text: angular

text: momenta

text: are

text: included

text: and

text: the

text: non-linear

text: parameter

text: κ

text: of

text: the

text: Coulomb

text: Sturmian

text: functions

text: is

text: equal

text: to

text: For

text: a

text: time

text: step

text: of

text: δt

text: a.u.,

text: we

text: compare

text: results

text: when

text: and

text: Krylov

text: vectors

text: are

text: used.

text: distribution

text: require

text: a

text: minimum

text: of

text: Krylov

text: vectors.

text: These

text: calculations

text: performed

text: in

text: a

text: Coulomb

text: Sturmian

text: basis

text: can

text: be

text: further

text: tested

text: by

text: varying

text: their

text: non-linear

text: pa-

text: rameter

text: κ.

text: If

text: instead

text: of

text: using

text: κ

text: we

text: use

text: κ

text: all

text: the

text: other

text: parameters

text: remaining

text: the

text: same,

text: we

text: again

text: obtain

text: a

text: completely

text: ﬂat

text: energy

text: distribution.

text: By

text: increasing

text: the

text: value

text: of

text: the

text: non-linear

text: parameter

text: κ,

text: the

text: value

text: of

text: the

text: eigenenergies

text: increases

text: thereby

text: increasing

text: the

text: stiﬀ

text: character

text: of

text: the

text: problem.

text: To

text: successfully

text: reproduce

text: an

text: accurate

text: energy

text: distribution

text: we

text: now

text: would

text: need

text: to

text: increase

text: the

text: number

text: of

text: Krylov

text: vectors.

text: If

text: on

text: the

text: other

text: hand,

text: we

text: keep

text: the

text: value

text: of

text: κ

text: equal

text: to

text: and

text: increase

text: the

text: number

text: of

text: basis

text: functions,

text: converged

text: results

text: are

text: only

text: obtained

text: when

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 2.5 Fatunla Arnoldi (N ) 2 krylov=25 u.) a. n (1.5 o uti b ri st 1 di y g r ne0.5 E 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Energy (a.u.) FIG. 12. (Color online) Energy distribution resulting from the interaction of the hydrogen atom with a cosine square pulse. Arnoldi’s propagator is used. The pulse has a peak intensity I=1014 Watt/cm2, a frequency ω = 0.114 a.u. and a duration of 20 optical cycles. We use a set of 600 Coulomb Sturmian functions per angular momentum. Ten angular momenta are taken into account and the non-linear parameter of the Coulomb Sturmian functions κ = 0.3. The Arnoldi propagator uses 25 Krylov − vectors and a time step of δt = 0.05 a.u. The relative diﬀerence between both curves is of the order of 10 3. 0.07 Arnoldi (N =70) 0.06 krylov a.u.)0.05 n ( utio0.04 b stri0.03 di y g0.02 r e n E 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Energy (a.u.) FIG. 13. (Color online)Energy distribution resulting from the interaction of the hydrogen atom with a cosine square pulse. Arnoldi’s propagator is used. The pulse has a peak intensity I=1014 Watt/cm2, a frequency ω = 0.0228 a.u. and a duration of 4 optical cycles. The basis-set of functions used is a set of 1200 Coulomb Sturmian functions per angular momentum. 80 angular momenta are taken into account and the non-linear parameter κ of the Coulomb Sturmian functions is equal to 0.3. The Arnoldi propagator uses 70 Krylov vectors and a time step of δt = 0.05 a.u. Krylov vectors are used. The increase in the number of Coulomb Sturmians generates higher eigenenergies thereby increasing again the stiﬀ character of the system. The eigenvalues of matrix h range from the eigenvalue of the initial state (by construction) to approximately the highest one of matrix H. In summary, any change which results in a higher maximum eigenvalue for H necessitates an increase in the number of Krylov subspace vectors required for convergence. It is interesting to note that to get an accurate spectrum, one of the eigenvalues of h must converge to 0.2 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: default

text: Fatunla

text: Arnoldi

text: (N

text: u.)

text: a.

text: n

text: o

text: uti

text: b

text: ri

text: st

text: di

text: y

text: g

text: r

text: E

text: Energy

text: (a.u.)

text: FIG.

text: (Color

label: caption

text: online)

text: Energy

text: distribution

text: resulting

text: from

text: the

text: interaction

text: of

text: the

text: hydrogen

text: atom

text: with

text: a

text: cosine

text: square

text: pulse.

text: Arnoldi’s

text: propagator

text: is

text: used.

text: The

text: pulse

text: has

text: a

text: peak

text: intensity

text: a

text: frequency

text: ω

text: a.u.

text: and

text: a

text: duration

text: of

text: optical

text: cycles.

text: We

text: use

text: a

text: set

text: of

text: Coulomb

text: Sturmian

text: functions

text: per

text: angular

text: momentum.

text: Ten

text: angular

text: momenta

text: are

text: taken

text: into

text: account

text: and

text: the

text: non-linear

text: parameter

text: of

text: the

text: Coulomb

text: Sturmian

text: functions

text: κ

text: The

text: Arnoldi

text: propagator

text: uses

text: Krylov

text: vectors

text: and

text: a

text: time

text: step

text: of

text: δt

text: a.u.

text: The

text: relative

text: diﬀerence

text: between

text: both

text: curves

text: is

text: of

text: the

text: order

text: of

text: Arnoldi

text: (N

text: krylov

text: n

text: b

text: di

text: y

text: r

text: e

text: n

text: E

text: Energy

text: (a.u.)

text: FIG.

text: (Color

text: online)Energy

text: distribution

text: resulting

text: from

text: the

text: interaction

text: of

text: the

text: hydrogen

text: atom

text: with

text: a

text: cosine

text: square

text: pulse.

text: Arnoldi’s

text: propagator

text: is

text: used.

text: The

text: pulse

text: has

text: a

text: peak

text: intensity

text: a

text: frequency

text: ω

text: a.u.

text: and

text: a

text: duration

text: of

text: optical

text: cycles.

text: The

text: basis-set

text: of

text: functions

text: used

text: is

text: a

text: set

text: of

text: Coulomb

text: Sturmian

text: functions

text: per

text: angular

text: momentum.

text: angular

text: momenta

text: are

text: taken

text: into

text: account

text: and

text: the

text: non-linear

text: parameter

text: κ

text: of

text: the

text: Coulomb

text: Sturmian

text: functions

text: is

text: equal

text: to

text: The

text: Arnoldi

text: propagator

text: uses

text: Krylov

text: vectors

text: and

text: a

text: time

text: step

text: of

text: δt

text: a.u.

text: Krylov

text: vectors

text: are

text: used.

text: The

text: increase

text: in

text: the

text: number

text: of

text: Coulomb

text: Sturmians

text: generates

text: higher

text: eigenenergies

text: thereby

text: increasing

text: again

text: the

text: stiﬀ

text: character

text: of

text: the

text: system.

text: The

text: eigenvalues

text: of

text: matrix

text: h

text: range

text: from

text: the

text: eigenvalue

text: of

text: the

text: initial

text: state

text: (by

text: construction)

text: to

text: approximately

text: the

text: highest

text: one

text: of

text: matrix

text: H.

text: In

text: summary,

text: any

text: change

text: which

text: results

text: in

text: a

text: higher

text: maximum

text: eigenvalue

text: for

text: H

text: necessitates

text: an

text: increase

text: in

text: the

text: number

text: of

text: Krylov

text: subspace

text: vectors

text: required

text: for

text: convergence.

text: It

text: is

text: interesting

text: to

text: note

text: that

text: to

text: get

text: an

text: accurate

text: spectrum,

text: one

text: of

text: the

text: eigenvalues

text: of

text: h

text: must

text: converge

text: to

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 inaccurately [15]. However, there is no mathematical tool which allows one to predict whether the numerical solution of a stiﬀ system will be accurate or not. Very often, the highest eigenvalues that correspond to very high energies do not play any physical role but, this does not imply that the error made in calculating the corresponding high energy components of the full numerical solution will not aﬀect the ﬁnal result. In fact, it is important to proceed on a case by case basis. In this contribution, we analyze in detail two explicit one-step integration schemes that have the required stability properties for dealing with stiﬀ systems. The ﬁrst method is due to Fatunla [16, 17] and the second one is a Krylov subspace method usually called the Arnoldi algorithm. The Arnoldi algorithm has already been used in many diﬀerent contexts: strong ﬁeld physics [18], condensed matter physics [13], etc... However, as far as we know, no systematic study of its stability and accuracy properties exists so far. In order to test the accuracy of both methods, we use a predictor-corrector scheme in which the predictor is either Fatunla’s method or Arnoldi’s algorithm while the corrector is a four-stage diagonally implicit Radau IIA method of order seven. Here, we consider the interaction of a quantum system with a strong and ultrashort electromagnetic pulse and test the three methods in the case of three diﬀerent quantum systems: a model potential, atomic hydrogen and helium. We also examine the performances of these explicit schemes in a completely diﬀerent context namely the calculation of a ﬁdelity function that measures the decoherence of two-electron quantum states when a time independent perturbation is applied to a planar two-electron quantum dot where both electrons are conﬁned in an anharmonic potential. In fact this is a diﬃcult problem, which exhibits a strong degree of stiﬀness although its Hamiltonian is time independent. Finally, let us mention that a comparison of diﬀerent time propagation algorithms for the time dependent Schr¨odinger equation may be found in [19]. This article is organized as follows. Section II is devoted to the general formulation of the TDSE. After some pre- liminary remarks, we give and discuss the general spectral representation of the TDSE and ﬁnally deﬁne the stability function of a given algorithm. In section III, we introduce the various algorithms (Fatunla’s method, Arnoldi’s algo- rithm and the predictor-corrector scheme) in the context of our model potential. For both explicit schemes, we give their stability function and analyze in detail their accuracy in the case of the model potential. Section IV is devoted to the results obtained with Fatunla’s and Arnoldi’s methods for the model potential, the interaction of atomic hydrogen with both a high and a very low frequency strong laser ﬁeld and single ionization of helium. Finally, we consider the problem of the planar two-electron quantum dot. Unless otherwise stated, atomic units are used throughout this paper. II. GENERAL FORMULATION OF THE TDSE A. Preliminary remarks Our aim is to study the interaction of a quantum system with an external time-dependent ﬁeld. Solving numerically the corresponding TDSE proceeds in two steps: the discretization in the spatial or/and energy domain of the equation and the time propagation of the solution. There are typically three ways of discretizing the TDSE: the ﬁnite diﬀerence grid (FDG) methods and the approaches based on spectral or ﬁnite element methods. The simplest approaches are the FDG methods. These methods based on a spatial discretization are essentially local. They are very often used because the subsequent time propagation involves solving very sparse systems of algebraic equations. However, it is often tricky to extract information on how these methods account for the electronic structure of the quantum system under consideration and some observables are sometimes diﬃcult to calculate. Furthermore, these methods yield ﬁnite-order rates of convergence in terms of the number of spatial grid points. In other words, the errors go to zero as the inverse of this number at a power given by the order of the method. The spectral methods based on an energy discretization are non-local. They consist in writing the solution as a L truncated expansion in terms of 2 integrable functions. These functions form a complete basis set. Diﬀerent choices of basis sets are possible, which usually depend on the physics of the problem. The most commonly used functions are the Hermite functions, the Coulomb Sturmian functions [20] and orthogonal polynomials. There are essentially two types of spectral methods: Galerkin and collocation [21]. In addition to the energy discretization as is the case in the Galerkin method, the collocation method involves also a spatial discretization. However, by contrast to the FDG methods, the grid mesh points are not arbitrarily chosen. They are the abscissae of the Gaussian quadrature associated with the basis functions. The spectral approaches are very appropriate for a very accurate description of the bound and resonance states of the quantum system under consideration. This is particularly true for resonance states very close to the ionization thresholds [22]. The convergence of the spectral methods in terms of the number of basis functions depends on the analytical properties of the solution. If the successive spatial derivatives of this solution do not exhibit singularities, the convergence is exponential. This means that the errors go to zero faster than any ﬁnite power of the inverse of the number of basis functions. On the other hand, if the solution exhibits singularities in

text: inaccurately

text: However,

text: there

text: is

text: no

text: mathematical

text: tool

text: which

text: allows

text: one

text: to

text: predict

text: whether

text: the

text: numerical

text: solution

text: of

text: a

text: stiﬀ

text: system

text: will

text: be

text: accurate

text: or

text: not.

text: Very

text: often,

text: the

text: highest

text: eigenvalues

text: that

text: correspond

text: to

text: very

text: high

text: energies

text: do

text: not

text: play

text: any

text: physical

text: role

text: but,

text: this

text: does

text: not

text: imply

text: that

text: the

text: error

text: made

text: in

text: calculating

text: the

text: corresponding

text: high

text: energy

text: components

text: of

text: the

text: full

text: numerical

text: solution

text: will

text: not

text: aﬀect

text: the

text: ﬁnal

text: result.

text: In

text: fact,

text: it

text: is

text: important

text: to

text: proceed

text: on

text: a

text: case

text: by

text: case

text: basis.

text: In

text: this

text: contribution,

text: we

text: analyze

text: in

text: detail

text: two

text: explicit

text: one-step

text: integration

text: schemes

text: that

text: have

text: the

text: required

text: stability

text: properties

text: for

text: dealing

text: with

text: stiﬀ

text: systems.

text: The

text: ﬁrst

text: method

text: is

text: due

text: to

text: Fatunla

text: and

text: the

text: second

text: one

text: is

text: a

text: Krylov

text: subspace

text: method

text: usually

text: called

text: the

text: Arnoldi

text: algorithm.

text: The

text: Arnoldi

text: algorithm

text: has

text: already

text: been

text: used

text: in

text: many

text: diﬀerent

text: contexts:

text: strong

text: ﬁeld

text: physics

text: condensed

text: matter

text: physics

text: etc...

text: However,

text: as

text: far

text: as

text: we

text: know,

text: no

text: systematic

text: study

text: of

text: its

text: stability

text: and

text: accuracy

text: properties

text: exists

text: so

text: far.

text: In

text: order

text: to

text: test

text: the

text: accuracy

text: of

text: both

text: methods,

text: we

text: use

text: a

text: predictor-corrector

text: scheme

text: in

text: which

text: the

text: predictor

text: is

text: either

text: Fatunla’s

text: method

text: or

text: Arnoldi’s

text: algorithm

text: while

text: the

text: corrector

text: is

text: a

text: four-stage

text: diagonally

text: implicit

text: Radau

text: IIA

text: method

text: of

text: order

text: seven.

text: Here,

text: we

text: consider

text: the

text: interaction

text: of

text: a

text: quantum

text: system

text: with

text: a

text: strong

text: and

text: ultrashort

text: electromagnetic

text: pulse

text: and

text: test

text: the

text: three

text: methods

text: in

text: the

text: case

text: of

text: three

text: diﬀerent

text: quantum

text: systems:

text: a

text: model

text: potential,

text: atomic

text: hydrogen

text: and

text: helium.

text: We

text: also

text: examine

text: the

text: performances

text: of

text: these

text: explicit

text: schemes

text: in

text: a

text: completely

text: diﬀerent

text: context

text: namely

text: the

text: calculation

text: of

text: a

text: ﬁdelity

text: function

text: that

text: measures

text: the

text: decoherence

text: of

text: two-electron

text: quantum

text: states

text: when

text: a

text: time

text: independent

text: perturbation

text: is

text: applied

text: to

text: a

text: planar

text: two-electron

text: quantum

text: dot

text: where

text: both

text: electrons

text: are

text: conﬁned

text: in

text: an

text: anharmonic

text: potential.

text: In

text: fact

text: this

text: is

text: a

text: diﬃcult

text: problem,

text: which

text: exhibits

text: a

text: strong

text: degree

text: of

text: stiﬀness

text: although

text: its

text: Hamiltonian

text: is

text: time

text: independent.

text: Finally,

text: let

text: us

text: mention

text: that

text: a

text: comparison

text: of

text: diﬀerent

text: time

text: propagation

text: algorithms

text: for

text: the

text: time

text: dependent

text: Schr¨odinger

text: equation

text: may

text: be

text: found

text: in

text: This

text: article

text: is

text: organized

text: as

text: follows.

text: Section

text: II

text: is

text: devoted

text: to

text: the

text: general

text: formulation

text: of

text: the

text: TDSE.

text: After

text: some

text: pre-

text: liminary

text: remarks,

text: we

text: give

text: and

text: discuss

text: the

text: general

text: spectral

text: representation

text: of

text: the

text: TDSE

text: and

text: ﬁnally

text: deﬁne

text: the

text: stability

text: function

text: of

text: a

text: given

text: algorithm.

text: In

text: section

text: III,

text: we

text: introduce

text: the

text: various

text: algorithms

text: (Fatunla’s

text: method,

text: Arnoldi’s

text: algo-

text: rithm

text: and

text: the

text: predictor-corrector

text: scheme)

text: in

text: the

text: context

text: of

text: our

text: model

text: potential.

text: For

text: both

text: explicit

text: schemes,

text: we

text: give

text: their

text: stability

text: function

text: and

text: analyze

text: in

text: detail

text: their

text: accuracy

text: in

text: the

text: case

text: of

text: the

text: model

text: potential.

text: Section

text: IV

text: is

text: devoted

text: to

text: the

text: results

text: obtained

text: with

text: Fatunla’s

text: and

text: Arnoldi’s

text: methods

text: for

text: the

text: model

text: potential,

text: the

text: interaction

text: of

text: atomic

text: hydrogen

text: with

text: both

text: a

text: high

text: and

text: a

text: very

text: low

text: frequency

text: strong

text: laser

text: ﬁeld

text: and

text: single

text: ionization

text: of

text: helium.

text: Finally,

text: we

text: consider

text: the

text: problem

text: of

text: the

text: planar

text: two-electron

text: quantum

text: dot.

text: Unless

text: otherwise

text: stated,

text: atomic

text: units

text: are

text: used

text: throughout

text: this

text: paper.

text: II.

text: GENERAL

text: FORMULATION

text: OF

text: THE

text: TDSE

text: A.

text: Preliminary

text: remarks

text: Our

text: aim

text: is

text: to

text: study

text: the

text: interaction

text: of

text: a

text: quantum

text: system

text: with

text: an

text: external

text: time-dependent

text: ﬁeld.

text: Solving

text: numerically

text: the

text: corresponding

text: TDSE

text: proceeds

text: in

text: two

text: steps:

text: the

text: discretization

text: in

text: the

text: spatial

text: or/and

text: energy

text: domain

text: of

text: the

text: equation

text: and

text: the

text: time

text: propagation

text: of

text: the

text: solution.

text: There

text: are

text: typically

text: three

text: ways

text: of

text: discretizing

text: the

text: TDSE:

text: the

text: ﬁnite

text: diﬀerence

text: grid

text: (FDG)

text: methods

text: and

text: the

text: approaches

text: based

text: on

text: spectral

text: or

text: ﬁnite

text: element

text: methods.

text: The

text: simplest

text: approaches

text: are

text: the

text: FDG

text: methods.

text: These

text: methods

text: based

text: on

text: a

text: spatial

text: discretization

text: are

text: essentially

text: local.

text: They

text: are

text: very

text: often

text: used

text: because

text: the

text: subsequent

text: time

text: propagation

text: involves

text: solving

text: very

text: sparse

text: systems

text: of

text: algebraic

text: equations.

text: However,

text: it

text: is

text: often

text: tricky

text: to

text: extract

text: information

text: on

text: how

text: these

text: methods

text: account

text: for

text: the

text: electronic

text: structure

text: of

text: the

text: quantum

text: system

text: under

text: consideration

text: and

text: some

text: observables

text: are

text: sometimes

text: diﬃcult

text: to

text: calculate.

text: Furthermore,

text: these

text: methods

text: yield

text: ﬁnite-order

text: rates

text: of

text: convergence

text: in

text: terms

text: of

text: the

text: number

text: of

text: spatial

text: grid

text: points.

text: In

text: other

text: words,

text: the

text: errors

text: go

text: to

text: zero

text: as

text: the

text: inverse

text: of

text: this

text: number

text: at

text: a

text: power

text: given

text: by

text: the

text: order

text: of

text: the

text: method.

text: The

text: spectral

text: methods

text: based

text: on

text: an

text: energy

text: discretization

text: are

text: non-local.

text: They

text: consist

text: in

text: writing

text: the

text: solution

text: as

text: a

text: L

text: truncated

text: expansion

text: in

text: terms

text: of

text: integrable

text: functions.

text: These

text: functions

text: form

text: a

text: complete

text: basis

text: set.

text: Diﬀerent

text: choices

text: of

text: basis

text: sets

text: are

text: possible,

text: which

text: usually

text: depend

text: on

text: the

text: physics

text: of

text: the

text: problem.

text: The

text: most

text: commonly

text: used

text: functions

text: are

text: the

text: Hermite

text: functions,

text: the

text: Coulomb

text: Sturmian

text: functions

text: and

text: orthogonal

text: polynomials.

text: There

text: are

text: essentially

text: two

text: types

text: of

text: spectral

text: methods:

text: Galerkin

text: and

text: collocation

text: In

text: addition

text: to

text: the

text: energy

text: discretization

text: as

text: is

text: the

text: case

text: in

text: the

text: Galerkin

text: method,

text: the

text: collocation

text: method

text: involves

text: also

text: a

text: spatial

text: discretization.

text: However,

text: by

text: contrast

text: to

text: the

text: FDG

text: methods,

text: the

text: grid

text: mesh

text: points

text: are

text: not

text: arbitrarily

text: chosen.

text: They

text: are

text: the

text: abscissae

text: of

text: the

text: Gaussian

text: quadrature

text: associated

text: with

text: the

text: basis

text: functions.

text: The

text: spectral

text: approaches

text: are

text: very

text: appropriate

text: for

text: a

text: very

text: accurate

text: description

text: of

text: the

text: bound

text: and

text: resonance

text: states

text: of

text: the

text: quantum

text: system

text: under

text: consideration.

text: This

text: is

text: particularly

text: true

text: for

text: resonance

text: states

text: very

text: close

text: to

text: the

text: ionization

text: thresholds

text: The

text: convergence

text: of

text: the

text: spectral

text: methods

text: in

text: terms

text: of

text: the

text: number

text: of

text: basis

text: functions

text: depends

text: on

text: the

text: analytical

text: properties

text: of

text: the

text: solution.

text: If

text: the

text: successive

text: spatial

text: derivatives

text: of

text: this

text: solution

text: do

text: not

text: exhibit

text: singularities,

text: the

text: convergence

text: is

text: exponential.

text: This

text: means

text: that

text: the

text: errors

text: go

text: to

text: zero

text: faster

text: than

text: any

text: ﬁnite

text: power

text: of

text: the

text: inverse

text: of

text: the

text: number

text: of

text: basis

text: functions.

text: On

text: the

text: other

text: hand,

text: if

text: the

text: solution

text: exhibits

text: singularities

text: in


text: 9 is above the upper bound. In all three cases, we observe a signiﬁcant loss of accuracy in ∆ at the very beginning of the time propagation. As described by Madron˜ero and Piraux [39], this is due to initially very small values of the denominators in Eq. (20) which leads to inaccurate values of the stiﬀness matrix elements and of the truncated error. This problem is therefore intrinsically related to Fatunla’s method and leads to diﬃculties in correctly controlling the time step. In fact , if we keep the time step constant from the beginning, we have a much better control of ∆. We have also checked that this is true even in the ﬁeld free case. On the other hand, in general, we see from the inset in Fig.2 that we maintain a higher accuracy when the constraint on the truncated error is more severe. In addition, we also observe several small jumps in ∆ the magnitude of which are much smaller than the jump in ∆ at the beginning of the time propagation. We attribute these jumps to an accumulation of roundoﬀ errors. Indeed, we expect more roundoﬀ errors in the case the constraint on the truncated error is the strongest since a smaller time step leads to a larger amount of calculations. Note that the jump observed in the red continuous line corresponds to a change of only one digit in the accuracy of the norm. The overall relative accuracy we obtain even for the most − severe constraint we use on the truncation error is of the order of 10 5. To achieve a greater accuracy, it is necessary to use a ﬁxed and very small time step. These results show clearly that the achievable accuracy for the adaptive time step approach in Fatunla’s method has a lower bound for a given initial time step. As a result, the use of Fatunla’s method rests on a compromise between the computer time required and the accuracy needed. In the following, we consider the interaction of helium with a strong laser pulse. In that case, the accuracy on the norm reduces to about 4 signiﬁcant digits when Fatunla’s method is used. This prevents us to calculate the probability of single ioniza- − tion in various channels where the latter one is less than 10 4 a.u. for ﬁeld intensities currently used in the experiments. In conclusion Fatunla’s method allows one to treat stiﬀ problems while fully exploiting the advantages of explicit schemes, namely that it only involves matrix vector multiplications. However it has its own limitations. B. Krylov subspace method In this section we consider a powerful method to propagate the TDSE solution, which provides accuracy of solutions and stability of propagation. It uses projection techniques on Krylov subspaces [40]. This approach was proposed by Arnoldi [41] in the calculation of the eigenstates of a matrix. Here we brieﬂy recall the method used by Arnoldi as a time propagator [42], to solve the diﬀerential equation (5). Since the overlap matrix is positive-deﬁnite, we can † use the Cholesky decomposition B = U U to form an orthonormal basis deﬁning the new coeﬃcients Φ = UΨ. The TDSE for these coeﬃcients is written in the form, (cid:54)100 m r o n 10−15 < |T | < 10−10 e h n+1 t 10−16 < |T | < 10−11 n n+1 or o 10−5 10−17 < |Tn+1| < 10−12 r r e 100 e t u ol 10−5 s b A 5 10−10 10−10 − −45 −40 −35 −30 0 1 0 50 100 150 Time (a.u.) | | | − | | | FIG. 2. (Color online) Absolute error in the norm ∆ = ( Ψ(r, t) Ψ(r, t ) ) on a logarithmic scale for diﬀerent lower and 0 upper bounds of the truncation error. The parameters of the Gaussian model problem are the same as in Fig.1. The inset is a blow-up of the region at the beginning of the time propagation. ##LTFigure##

text: is

text: above

text: the

text: upper

text: bound.

text: In

text: all

text: three

text: cases,

text: we

text: observe

text: a

text: signiﬁcant

text: loss

text: of

text: accuracy

text: in

text: at

text: the

text: very

text: beginning

text: of

text: the

text: time

text: propagation.

text: As

text: described

text: by

text: Madron˜ero

text: and

text: Piraux

text: this

text: is

text: due

text: to

text: initially

text: very

text: small

text: values

text: of

text: the

text: denominators

text: in

text: Eq.

text: which

text: leads

text: to

text: inaccurate

text: values

text: of

text: the

text: stiﬀness

text: matrix

text: elements

text: and

text: of

text: the

text: truncated

text: error.

text: This

text: problem

text: is

text: therefore

text: intrinsically

text: related

text: to

text: Fatunla’s

text: method

text: and

text: leads

text: to

text: diﬃculties

text: in

text: correctly

text: controlling

text: the

text: time

text: step.

text: In

text: fact

text: if

text: we

text: keep

text: the

text: time

text: step

text: constant

text: from

text: the

text: beginning,

text: we

text: have

text: a

text: much

text: better

text: control

text: of

text: We

text: have

text: also

text: checked

text: that

text: this

text: is

text: true

text: even

text: in

text: the

text: ﬁeld

text: free

text: case.

text: On

text: the

text: other

text: hand,

text: in

text: general,

text: we

text: see

text: from

text: the

text: inset

text: in

text: that

text: we

text: maintain

text: a

text: higher

text: accuracy

text: when

text: the

text: constraint

text: on

text: the

text: truncated

text: error

text: is

text: more

text: severe.

text: In

text: addition,

text: we

text: also

text: observe

text: several

text: small

text: jumps

text: in

text: the

text: magnitude

text: of

text: which

text: are

text: much

text: smaller

text: than

text: the

text: jump

text: in

text: at

text: the

text: beginning

text: of

text: the

text: time

text: propagation.

text: We

text: attribute

text: these

text: jumps

text: to

text: an

text: accumulation

text: of

text: roundoﬀ

text: errors.

text: Indeed,

text: we

text: expect

text: more

text: roundoﬀ

text: errors

text: in

text: the

text: case

text: the

text: constraint

text: on

text: the

text: truncated

text: error

text: is

text: the

text: strongest

text: since

text: a

text: smaller

text: time

text: step

text: leads

text: to

text: a

text: larger

text: amount

text: of

text: calculations.

text: Note

text: that

text: the

text: jump

text: observed

text: in

text: the

text: red

text: continuous

text: line

text: corresponds

text: to

text: a

text: change

text: of

text: only

text: one

text: digit

text: in

text: the

text: accuracy

text: of

text: the

text: norm.

text: The

text: overall

text: relative

text: accuracy

text: we

text: obtain

text: even

text: for

text: the

text: most

text: severe

text: constraint

text: we

text: use

text: on

text: the

text: truncation

text: error

text: is

text: of

text: the

text: order

text: of

text: To

text: achieve

text: a

text: greater

text: accuracy,

text: it

text: is

text: necessary

text: to

text: use

text: a

text: ﬁxed

text: and

text: very

text: small

text: time

text: step.

text: These

text: results

text: show

text: clearly

text: that

text: the

text: achievable

text: accuracy

text: for

text: the

text: adaptive

text: time

text: step

text: approach

text: in

text: Fatunla’s

text: method

text: has

text: a

text: lower

text: bound

text: for

text: a

text: given

text: initial

text: time

text: step.

text: As

text: a

text: result,

text: the

text: use

text: of

text: Fatunla’s

text: method

text: rests

text: on

text: a

text: compromise

text: between

text: the

text: computer

text: time

text: required

text: and

text: the

text: accuracy

text: needed.

text: In

text: the

text: following,

text: we

text: consider

text: the

text: interaction

text: of

text: helium

text: with

text: a

text: strong

text: laser

text: pulse.

text: In

text: that

text: case,

text: the

text: accuracy

text: on

text: the

text: norm

text: reduces

text: to

text: about

text: signiﬁcant

text: digits

text: when

text: Fatunla’s

text: method

text: is

text: used.

text: This

text: prevents

text: us

text: to

text: calculate

text: the

text: probability

text: of

text: single

text: ioniza-

text: tion

text: in

text: various

text: channels

text: where

text: the

text: latter

text: one

text: is

text: less

text: than

text: a.u.

text: for

text: ﬁeld

text: intensities

text: currently

text: used

text: in

text: the

text: experiments.

text: In

text: conclusion

text: Fatunla’s

text: method

text: allows

text: one

text: to

text: treat

text: stiﬀ

text: problems

text: while

text: fully

text: exploiting

text: the

text: advantages

text: of

text: explicit

text: schemes,

text: namely

text: that

text: it

text: only

text: involves

text: matrix

text: vector

text: multiplications.

text: However

text: it

text: has

text: its

text: own

text: limitations.

text: B.

text: Krylov

text: subspace

text: method

text: In

text: this

text: section

text: we

text: consider

text: a

text: powerful

text: method

text: to

text: propagate

text: the

text: TDSE

text: solution,

text: which

text: provides

text: accuracy

text: of

text: solutions

text: and

text: stability

text: of

text: propagation.

text: It

text: uses

text: projection

text: techniques

text: on

text: Krylov

text: subspaces

text: This

text: approach

text: was

text: proposed

text: by

text: Arnoldi

text: in

text: the

text: calculation

text: of

text: the

text: eigenstates

text: of

text: a

text: matrix.

text: Here

text: we

text: brieﬂy

text: recall

text: the

text: method

text: used

text: by

text: Arnoldi

text: as

text: a

text: time

text: propagator

text: to

text: solve

text: the

text: diﬀerential

text: equation

text: Since

text: the

text: overlap

text: matrix

text: is

text: positive-deﬁnite,

text: we

text: can

text: use

text: the

text: Cholesky

text: decomposition

text: B

text: U

text: U

text: to

text: form

text: an

text: orthonormal

text: basis

text: deﬁning

text: the

text: new

text: coeﬃcients

text: Φ

text: UΨ.

text: The

text: TDSE

text: for

text: these

text: coeﬃcients

text: is

text: written

text: in

text: the

text: form,

text: m

text: r

text: o

text: n

text: |T

text: e

text: h

text: t

text: |T

text: n

text: or

text: o

text: r

text: r

text: e

text: e

text: t

text: u

text: ol

text: s

text: b

text: A

text: Time

text: (a.u.)

text: FIG.

text: (Color

text: online)

text: Absolute

text: error

text: in

text: the

text: norm

text: Ψ(r,

text: t)

text: Ψ(r,

text: t

text: on

text: a

text: logarithmic

text: scale

text: for

text: diﬀerent

text: lower

text: and

text: upper

text: bounds

text: of

text: the

text: truncation

text: error.

text: The

text: parameters

text: of

text: the

text: Gaussian

text: model

text: problem

text: are

text: the

text: same

text: as

text: in

text: The

text: inset

text: is

text: a

text: blow-up

text: of

text: the

text: region

text: at

text: the

text: beginning

text: of

text: the

text: time

text: propagation.

text: ##LTFigure##


text: Here, M − − 0 = A (1)(715 8959a + 50042a2 142196a3 + 169032a4 + 26688a5 NLO 0 − − − − 150912a6)/(27a(1 9a + 26a2 24a3)) + 2A (2)(46 1015a + 6578a2 0 − − − − 18023a3 + 17406a4 + 8688a5 18720a6)/(27a(1 9a + 26a2 24a3)) − − − − − ((16n 177) 8(8n 91)a + 12(16n 151)a2)m2B (1)/9 + (2 l l l b 0 − − − +31a 824a2 + 2172a3)m2B (2)/(9a) 11(1 4a + 12a2)m2B (3)/3 b 0 b 0 − − − − − +2(11 11a + 48a2 180a3)m2B (4)/(3(1 3a)) + 2(1 18a + 62a2 152a3 b 0 − − − − +168a4 + 864a5 1536a6)m2B (5)/(9a(1 6a + 8a2)) 2(46 851a + 5040a2 b 0 − − − 11755a3 + 6560a4 + 4248a5 + 17592a6 29664a7)m2B (6)/(27a(1 9a b 0 − − − − +26a2 24a3)) 2(2 11a 326a2 + 738a3 + 444a4)m2B (7)/(9a) b 0 − − − 6a(11 + 6a)m4C (1) 4a(19 52a + 60a2)m4C (2)/9 + 8a(130a b 0 b 0 − − − − − 79)m4C (3)/9 2(20 207a + 242a2 24a3)m4C (4)/9 4(1 + 11a b 0 b 0 − − − +12a2 12a3)m4C (5)/9 8(15 89a + 130a2 + 54a3)m4C (6)/9 b 0 b 0 − − − +8(4 107a + 102a2)m4C (7)/9 + 16(5 14a 8a2 + 48a3)m4C (8)/9 b 0 b 0 − m2 − m2 +8(1 4a + 12a2)m2 ln( b )/3 + 32(1 4a + 12a2)m2 ln( c )/3 b µ2 b µ2 − − − − +2(8n 561 + (6693 88n )a + 2(208n 17631)a2 + 6(15947 176n )a3 l l l l − − +48(24n 2383)a4 + 23904a5)m2/(27(1 7a + 12a2)) (A5) l b 11 ##LTLine## ##LTLine##

text: Here,

text: M

text: A

text: NLO

text: l

text: l

text: l

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: ln(

text: b

text: ln(

text: c

text: b

text: µ2

text: b

text: µ2

text: )a

text: l

text: l

text: l

text: l

text: l

text: b

text: ##LTLine##

text: ##LTLine##


text: µ2 B (2) = B (4m2, 0, 0) = 2 + ln( ) + iπ , (A11) 0 0 b 4m2 b µ2 B (3) = B (m2, 0, m2) = 2 + ln( ) , (A12) 0 0 b b m2 b µ2 − √π B (4) = B (m2, m2, m2) = ln( ) + 2 , (A13) 0 0 b b b m2 (cid:88) 3 b µ2 − 2 B (5) = B (m2, m2, m2) = 2 + ln( ) 2x ln(x ) + iπ , (A14) 0 0 b c c m2 i i (cid:88)b i=1 − µ2 − 2 x 1 − − B (6) = B (m2, m2, m2) = 2 + ln( ) (x ln( i ) ln(x 1)) , (A15) 0 0 c b c m2 i x i c i=1 i µ2 − 2 − B (7) = B (2m2 + m2, 0, m2) = 2 + ln( ) (ln(2a) iπ) , (A16) 0 0 b c c m2 2 + a c C (1) = C (m2, m2, m2, m2, m2, m2) , (A17) 0 0 b c c b b c C (2) = C (m2, m2, 2m2 + m2, m2, m2, 0) , (A18) 0 0 b c b c c c C (3) = C (4m2, m2, 2m2 + m2, 0, 0, m2) , (A19) 0 0 b c b c c C (4) = C (m2, m2, m2, 0, 0, m2) , (A20) 0 0 b c c c C (5) = C (m2, m2, m2, m2, m2, m2) , (A21) 0 0 b c c c c b C (6) = C (m2, m2, 2m2 + m2, 0, 0, m2) , (A22) 0 0 b c b c c C (7) = C (m2, m2, 2m2 + m2, 0, m2, m2) , (A23) 0 0 b c b c b c C (8) = C (m2, 4m2, 2m2 + m2, 0, m2, m2) . (A24) 0 0 c c b c c c In above expressions, the divergent parts of the A and B functions have been 0 0 removed, the value of C function was given in [? ] and we rechecked it analytically. 0 The numerical evaluation was performed by means of the software LoopTools. 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: µ2

text: B

text: B

text: ln(

text: iπ

text: b

text: b

text: µ2

text: B

text: B

text: ln(

text: b

text: b

text: b

text: µ2

text: √π

text: B

text: B

text: ln(

text: b

text: b

text: b

text: b

text: µ2

text: B

text: B

text: ln(

text: ln(x

text: iπ

text: b

text: c

text: c

text: i

text: i

text: µ2

text: x

text: B

text: B

text: ln(

text: (x

text: ln(

text: i

text: ln(x

text: c

text: b

text: c

text: i

text: x

text: i

text: c

text: i

text: µ2

text: B

text: B

text: ln(

text: iπ)

text: b

text: c

text: c

text: a

text: c

text: C

text: C

text: b

text: c

text: c

text: b

text: b

text: c

text: C

text: C

text: b

text: c

text: b

text: c

text: c

text: c

text: C

text: C

text: b

text: c

text: b

text: c

text: c

text: C

text: C

text: b

text: c

text: c

text: c

text: C

text: C

text: b

text: c

text: c

text: c

text: c

text: b

text: C

text: C

text: b

text: c

text: b

text: c

text: c

text: C

text: C

text: b

text: c

text: b

text: c

text: b

text: c

text: C

text: C

text: c

text: c

text: b

text: c

text: c

text: c

text: In

text: above

text: expressions,

text: the

text: divergent

text: parts

text: of

text: the

text: A

text: and

text: B

text: functions

text: have

text: been

text: removed,

text: the

text: value

text: of

text: C

text: function

text: was

text: given

text: in

text: and

text: we

text: rechecked

text: it

text: analytically.

text: The

text: numerical

text: evaluation

text: was

text: performed

text: by

text: means

text: of

text: the

text: software

text: LoopTools.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: conclusions. Several formulas are provided in the Appendix for reference. II. CALCULATION SCHEME DESCRIPTION AND FORMALISM → FIG. 1: Typical Feynman diagrams for χ J/ψJ/ψ decays. bJ In the calculation, we use Mathematica package FeynArts [? ] to generate all the Feynman diagrams and decay amplitudes. Certain typical representative diagrams are shown in Fig.1. All the projection operators for quarkonium in this work may be found (cid:18) (cid:19) in Ref.[? ? ]. For the spin-triplet color-singlet state, the matrix product is expressed as: √ 1 (cid:54) − (cid:54) ∗ (cid:54) (cid:54) ⊗ √1 c v(p¯)u¯(p) = (p¯ m ) (cid:15) (P + 2E)(p + m ) . (1) 4 2E(E + m) c S c N c − (cid:54) ∗ Here p = P + q, p¯ = P q respectively are the momenta of quark and antiquark, (cid:15) is 2 2 − a spin polarization vector, E2 = P 2/4 = m2 q2, N = 3, and 1 represents the unit c c c color matrix. For the spin-singlet and color-singlet state, the projection operator may ∗ be obtained by replacing the (cid:15) in (??) with a γ . S 5 Using the method described in [? ] and applying FeynCalc [? ] to the calculation of amplitudes, the tree-level results are readily obtained. For One-loop QCD corrections, more operations are necessary before arriving at the ﬁnal result. For χ and χ decays b0 b2 3 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: conclusions.

text: Several

text: formulas

text: are

text: provided

text: in

text: the

text: Appendix

text: for

text: reference.

text: II.

label: section

text: CALCULATION

text: SCHEME

text: DESCRIPTION

text: AND

text: FORMALISM

text: FIG.

text: Typical

text: Feynman

text: diagrams

text: for

text: χ

text: J/ψJ/ψ

text: decays.

text: bJ

text: In

text: the

text: calculation,

text: we

text: use

text: Mathematica

text: package

text: FeynArts

text: to

text: generate

text: all

text: the

text: Feynman

text: diagrams

text: and

text: decay

text: amplitudes.

text: Certain

text: typical

text: representative

text: diagrams

text: are

text: shown

text: in

text: All

text: the

text: projection

text: operators

text: for

text: quarkonium

text: in

text: this

text: work

text: may

text: be

text: found

text: in

text: Ref.[?

text: For

text: the

text: spin-triplet

text: color-singlet

text: state,

text: the

text: matrix

text: product

text: is

text: expressed

text: as:

text: √1

text: c

text: v(p¯)u¯(p)

text: (p¯

text: m

text: (P

text: m

text: m)

text: c

text: S

text: c

text: N

text: c

text: Here

text: p

text: P

text: q,

text: p¯

text: P

text: q

text: respectively

text: are

text: the

text: momenta

text: of

text: quark

text: and

text: antiquark,

text: is

text: a

text: spin

text: polarization

text: vector,

text: P

text: N

text: and

text: represents

text: the

text: unit

text: c

text: c

text: c

text: color

text: matrix.

text: For

text: the

text: spin-singlet

text: and

text: color-singlet

text: state,

text: the

text: projection

text: operator

text: may

text: be

text: obtained

text: by

text: replacing

text: the

text: in

text: with

text: a

text: γ

text: S

text: Using

text: the

text: method

text: described

text: in

text: and

text: applying

text: FeynCalc

text: to

text: the

text: calculation

text: of

text: amplitudes,

text: the

text: tree-level

text: results

text: are

text: readily

text: obtained.

text: For

text: One-loop

text: QCD

text: corrections,

text: more

text: operations

text: are

text: necessary

text: before

text: arriving

text: at

text: the

text: ﬁnal

text: result.

text: For

text: χ

text: and

text: χ

text: decays

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: On the other hand, the numerical scheme (2.11) is mass conservative at the discrete level, ≡ provided that ψ0 0: (cid:88) − N 1 ≡ uk := h uk u0 = C¯ . (4.5) i 0 i=0 ∈ B ≥ Meanwhile, for Uk N, for any k 0, we observe that N (cid:90) (cid:90) · ≡ · Uk = U ( , tk) dx U ( , 0) = U0. (4.6) N N Ω Ω As a result, we arrive at an O(hm) order average for the numerical error function at each time step: − − − ∀ ≥ u˜k = Uk uk = Uk uk = U0 u0 = O(hm), k 0, (4.7) which comes from the error associated with the projection. This is equivalent to (cid:90) ∀ ≥ u˜k dx = u˜k = O(hm), k 0, (4.8) N Ω ∈ B with the ﬁrst step based on the fact that u˜k N. As an application of elliptic regularity, N we arrive at (cid:18) (cid:90) (cid:19) (cid:13) (cid:13) (cid:0)(cid:13) (cid:13) (cid:1) (cid:13) (cid:13) (cid:13) (cid:13) (cid:107) (cid:107) ≤ ≤ u˜k C ∂2u˜k + u˜k dx C D2 u˜k + hm , (4.9) N H2 x N L2 N N 2 Ω ∈ B in which the fact that u˜k N was used in the last step. This ﬁnishes the proof of Lemma N 4.1. Meanwhile, for a semi-discrete function w (continuous in space and discrete in time), the following norms are deﬁned: (cid:13) (cid:13) (cid:13) (cid:13) (cid:107) (cid:107) ≥ w (cid:96)∞(0,T∗;Hk) = 0≤mka≤xK wk Hk , for any integer k 0 . (4.10) Finally, we state the main result of this paper: 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: On

text: the

text: other

text: hand,

text: the

text: numerical

text: scheme

text: is

text: mass

text: conservative

text: at

text: the

text: discrete

text: level,

text: provided

text: that

text: ψ0

label: equation

text: N

text: uk

text: h

text: uk

text: C¯

text: i

text: B

text: Meanwhile,

text: for

text: Uk

text: N,

text: for

text: any

text: k

text: we

text: observe

text: that

text: N

text: Uk

text: U

text: tk)

text: dx

text: U

text: N

text: N

text: Ω

text: Ω

text: As

text: a

text: result,

text: we

text: arrive

text: at

text: an

text: O(hm)

text: order

text: average

text: for

text: the

text: numerical

text: error

text: function

text: at

text: each

text: time

text: step:

text: u˜k

text: Uk

text: uk

text: Uk

text: uk

text: O(hm),

text: k

text: which

text: comes

text: from

text: the

text: error

text: associated

text: with

text: the

text: projection.

text: This

text: is

text: equivalent

text: to

text: u˜k

text: dx

text: u˜k

text: O(hm),

text: k

text: N

text: Ω

text: B

text: with

text: the

text: ﬁrst

text: step

text: based

text: on

text: the

text: fact

text: that

text: u˜k

text: N.

text: As

text: an

text: application

text: of

text: elliptic

text: regularity,

text: N

text: we

text: arrive

text: at

text: u˜k

text: C

text: u˜k

text: dx

text: C

text: u˜k

text: hm

text: N

text: x

text: N

text: N

text: N

text: Ω

text: B

text: in

text: which

text: the

text: fact

text: that

text: u˜k

text: N

text: was

text: used

text: in

text: the

text: last

text: step.

text: This

text: ﬁnishes

text: the

text: proof

text: of

text: Lemma

text: N

text: Meanwhile,

text: for

text: a

text: semi-discrete

text: function

text: w

text: (continuous

text: in

text: space

text: and

text: discrete

text: in

text: time),

text: the

text: following

text: norms

text: are

text: deﬁned:

text: w

text: wk

text: Hk

text: for

text: any

text: integer

text: k

text: Finally,

text: we

text: state

text: the

text: main

text: result

text: of

text: this

text: paper:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Theorem 4.2. For any ﬁnal time T > 0, assume the exact solution u to the GB equation e (1.1) given by (3.21). Denote u as the continuous (in space) extension of the fully discrete ∆t,h → numerical solution given by scheme 2.11. As ∆t, h 0, the following convergence result is valid: (cid:0) (cid:1) (cid:107) − (cid:107) (cid:107) − (cid:107) ≤ u∆t,h ue (cid:96)∞(0,T∗;H2) + ψ∆t,h ψe (cid:96)∞(0,T∗;L2) C ∆t2 + hm , (4.11) provided that the time step ∆t and the space grid size h are bounded by given constants which are only dependent on the exact solution. Note that the convergence constant in (4.11) also depend on the exact solution as well as T. Proof. Subtracting (2.11) from (3.22) yields − ψ˜n+1 ψ˜n −1 (cid:32) 1 (cid:33) = D4 (u˜n+1(cid:88)+ u˜n) + D2 (u˜n+1 + u˜n) +(cid:88)τn ∆t 2 N 2 N 1 − − p 1 p 1 3 − − − 1 − − − − − +D2 u˜n (Un)k(un)p 1 k u˜n 1 (Un 1)k(un 1)p 1 k ,(4.12) N 2 2 k=0 k=0 − u˜n+1 u˜n ψ˜n+1 + ψ˜n = + ∆tτn. (4.13) ∆t 2 2 ∞ Also note a W2, bound for the constructed approximate solution (cid:107) (cid:107) ≤ ∗ (cid:107) (cid:107) ≤ ∗ (cid:107) (cid:107) ≤ ∗ (cid:107) (cid:107) ≤ ∗ UN L∞(0,T∗;W2,∞) C , i.e. UNn L∞ C , (UN)nx L∞ C , (UN)nxx L∞ C , (4.14) ≥ for any n 0, which comes from the regularity of the constructed solution. An a-priori H2 assumption up to time step tn. We assume a-priori that the numerical − error function (for u) h(cid:13)as a(cid:13)n H2 bound at time steps tn, tn 1, (cid:13) (cid:13) ≤ I − u˜k 1, with u˜k = u˜k, for k = n, n 1, (4.15) N H2 N N ∞ so that the H2 and W1, bound for the numerical solution (up to tn) is available (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)− (cid:13) ≤ ≤ ∗ uk = Uk u˜k Uk + u˜k C + 1 := C˜ , N H2 N N H2 N H2 N H2 0 ≤ ≤ ukN W1,∞ C ukN H2 CC˜0 := C˜1, (4.16) 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Theorem

text: For

text: any

text: ﬁnal

text: time

text: T

text: assume

text: the

text: exact

text: solution

text: u

text: to

text: the

text: GB

text: equation

text: e

text: given

text: by

text: Denote

text: u

text: as

text: the

text: continuous

text: (in

text: space)

text: extension

text: of

text: the

text: fully

text: discrete

text: ∆t,h

text: numerical

text: solution

text: given

text: by

text: scheme

text: As

text: ∆t,

text: h

text: the

text: following

text: convergence

text: result

text: is

text: valid:

text: u∆t,h

text: ue

text: ψ∆t,h

text: ψe

text: C

text: hm

text: provided

text: that

text: the

text: time

text: step

text: ∆t

text: and

text: the

text: space

text: grid

text: size

text: h

text: are

text: bounded

text: by

text: given

text: constants

text: which

text: are

text: only

text: dependent

text: on

text: the

text: exact

text: solution.

text: Note

text: that

text: the

text: convergence

text: constant

text: in

text: also

text: depend

text: on

text: the

text: exact

text: solution

text: as

text: well

text: as

text: T.

text: Proof.

text: Subtracting

text: from

text: yields

text: ψ˜n

text: −1

text: u˜n)

text: u˜n)

text: ∆t

text: N

text: N

text: p

text: p

text: u˜n

text: (Un)k(un)p

text: k

text: u˜n

text: (Un

text: k

text: N

text: u˜n

text: ψ˜n

text: ∆tτn.

text: ∆t

text: Also

text: note

text: a

text: bound

text: for

text: the

text: constructed

text: approximate

text: solution

text: UN

text: C

text: i.e.

text: UNn

text: L∞

text: C

text: (UN)nx

text: L∞

text: C

text: (UN)nxx

text: L∞

text: C

text: for

text: any

text: n

text: which

text: comes

text: from

text: the

text: regularity

text: of

text: the

text: constructed

text: solution.

text: An

text: a-priori

text: assumption

text: up

text: to

text: time

text: step

text: tn.

text: We

text: assume

text: a-priori

text: that

text: the

text: numerical

text: error

text: function

text: (for

text: u)

text: bound

text: at

text: time

text: steps

text: tn,

text: tn

text: I

text: u˜k

text: with

text: u˜k

text: u˜k,

text: for

text: k

text: n,

text: n

text: N

text: N

text: N

text: so

text: that

text: the

text: and

text: bound

text: for

text: the

text: numerical

text: solution

text: (up

text: to

text: tn)

text: is

text: available

text: uk

text: Uk

text: u˜k

text: Uk

text: u˜k

text: C

text: C˜

text: N

text: N

text: N

text: N

text: N

text: ukN

text: C

text: ukN

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: that in [17]. To overcome the diﬃculty associated with the second order temporal derivative in the hyperbolic equation, we introduce a new variable ψ to approximate u , which greatly facilitates t the numerical implementation. On the other hand, the corresponding second order consistency analysis becomes non-trivial because of an O(∆t2) numerical error between the centered diﬀerence of u and the mid-point average of ψ. Without a careful treatment, such an O(∆t2) numerical error might seem to introduce a reduction of temporal accuracy, because of the second order time derivative involved in the equation. To overcome this diﬃculty, we perform a higher order consistency analysis by an asymptotic expansion; as a result, the constructed approximate solution satisﬁes the numerical scheme with a higher order truncation error. Furthermore, a projection of the exact solution onto the Fourier space leads to an optimal regularity requirement. For the nonlinear stability and convergence analysis, we have to obtain a direct estimate of the (discrete) H2 norm of the nonlinear numerical error function. This estimate relies on the aliasing error control lemma for pseudospectral approximation to nonlinear terms, which was proven in a recent work [19]. That’s the key reason why we are able to overcome the key diﬃculty in the ∞ ∗ ∞ ∗ nonlinear estimate and obtain an (cid:96) (0, T ; H2) convergence for u and (cid:96) (0, T ; (cid:96)2) convergence for u . We prove that the proposed numerical scheme is fully consistent (with a higher order t ∗ expansion), stable and convergent in the H2 norm up to some ﬁxed ﬁnal time T . In turn, the maximum norm bound of the numerical solution is automatically obtained, because of the H2 error estimate and the corresponding Sobolev embedding. Therefore, the inverse inequality in the stability analysis is not needed and any scaling law between ∆t and h is avoided, compared ≤ with the ∆t Ch2 constraint reported in [17]. This paper is outlined as follows. In Section 2 we review the Fourier spectral and pseu- dospectral diﬀerentiation, recall an aliasing error control lemma (proven in [19]), and present an alternate second order (in time) pseudospectral scheme for the GB equation (1.1). In Section 3, the consistency analysis of the scheme is studied in detail. The stability and convergence anal- ysis is reported in Section 4. A simple numerical result is presented in Section 5. Finally, some concluding remarks are made in Section 6. 2 The Numerical Scheme 2.1 Review of Fourier spectral and pseudospectral approximations ∈ For f(x) L2(Ω), Ω = (0, L), with Fourier series (cid:90) (cid:88) ∞ − f(x) = fˆe2πilx/L, with fˆ = f(x)e 2πilx/Ldx, (2.2) l l l=−∞ Ω B its truncated series is deﬁned as the projection onto the space N of trigonometric polyno- mials in x of degree up to N, given by (cid:88) P N f(x) = fˆe2πilx/L. (2.3) N l − l= N 3

text: that

text: in

text: To

text: overcome

text: the

text: diﬃculty

text: associated

text: with

text: the

text: second

text: order

text: temporal

text: derivative

text: in

text: the

text: hyperbolic

text: equation,

text: we

text: introduce

text: a

text: new

text: variable

text: ψ

text: to

text: approximate

text: u

text: which

text: greatly

text: facilitates

text: t

text: the

text: numerical

text: implementation.

text: On

text: the

text: other

text: hand,

text: the

text: corresponding

text: second

text: order

text: consistency

text: analysis

text: becomes

text: non-trivial

text: because

text: of

text: an

text: numerical

text: error

text: between

text: the

text: centered

text: diﬀerence

text: of

text: u

text: and

text: the

text: mid-point

text: average

text: of

text: ψ.

text: Without

text: a

text: careful

text: treatment,

text: such

text: an

text: numerical

text: error

text: might

text: seem

text: to

text: introduce

text: a

text: reduction

text: of

text: temporal

text: accuracy,

text: because

text: of

text: the

text: second

text: order

text: time

text: derivative

text: involved

text: in

text: the

text: equation.

text: To

text: overcome

text: this

text: diﬃculty,

text: we

text: perform

text: a

text: higher

text: order

text: consistency

text: analysis

text: by

text: an

text: asymptotic

text: expansion;

text: as

text: a

text: result,

text: the

text: constructed

text: approximate

text: solution

text: satisﬁes

text: the

text: numerical

text: scheme

text: with

text: a

text: higher

text: order

text: truncation

text: error.

text: Furthermore,

text: a

text: projection

text: of

text: the

text: exact

text: solution

text: onto

text: the

text: Fourier

text: space

text: leads

text: to

text: an

text: optimal

text: regularity

text: requirement.

text: For

text: the

text: nonlinear

text: stability

text: and

text: convergence

text: analysis,

text: we

text: have

text: to

text: obtain

text: a

text: direct

text: estimate

text: of

text: the

text: (discrete)

text: norm

text: of

text: the

text: nonlinear

text: numerical

text: error

text: function.

text: This

text: estimate

text: relies

text: on

text: the

text: aliasing

text: error

text: control

text: lemma

text: for

text: pseudospectral

text: approximation

text: to

text: nonlinear

text: terms,

text: which

text: was

text: proven

text: in

text: a

text: recent

text: work

text: That’s

text: the

text: key

text: reason

text: why

text: we

text: are

text: able

text: to

text: overcome

text: the

text: key

text: diﬃculty

text: in

text: the

text: nonlinear

text: estimate

text: and

text: obtain

text: an

text: T

text: convergence

text: for

text: u

text: and

text: T

text: convergence

text: for

text: u

text: We

text: prove

text: that

text: the

text: proposed

text: numerical

text: scheme

text: is

text: fully

text: consistent

text: (with

text: a

text: higher

text: order

text: t

text: expansion),

text: stable

text: and

text: convergent

text: in

text: the

text: norm

text: up

text: to

text: some

text: ﬁxed

text: ﬁnal

text: time

text: T

text: In

text: turn,

text: the

text: maximum

text: norm

text: bound

text: of

text: the

text: numerical

text: solution

text: is

text: automatically

text: obtained,

text: because

text: of

text: the

text: error

text: estimate

text: and

text: the

text: corresponding

text: Sobolev

text: embedding.

text: Therefore,

text: the

text: inverse

text: inequality

text: in

text: the

text: stability

text: analysis

text: is

text: not

text: needed

text: and

text: any

text: scaling

text: law

text: between

text: ∆t

text: and

text: h

text: is

text: avoided,

text: compared

text: with

text: the

text: ∆t

text: constraint

text: reported

text: in

text: This

text: paper

text: is

text: outlined

text: as

text: follows.

text: In

text: Section

text: we

text: review

text: the

text: Fourier

text: spectral

text: and

text: pseu-

text: dospectral

text: diﬀerentiation,

text: recall

text: an

text: aliasing

text: error

text: control

text: lemma

text: (proven

text: in

text: and

text: present

text: an

text: alternate

text: second

text: order

text: (in

text: time)

text: pseudospectral

text: scheme

text: for

text: the

text: GB

text: equation

text: In

text: Section

text: the

text: consistency

text: analysis

text: of

text: the

text: scheme

text: is

text: studied

text: in

text: detail.

text: The

text: stability

text: and

text: convergence

text: anal-

text: ysis

text: is

text: reported

text: in

text: Section

text: A

text: simple

text: numerical

text: result

text: is

text: presented

text: in

text: Section

text: Finally,

text: some

text: concluding

text: remarks

text: are

text: made

text: in

text: Section

text: The

text: Numerical

text: Scheme

text: Review

text: of

text: Fourier

text: spectral

text: and

text: pseudospectral

text: approximations

text: For

text: f(x)

text: Ω

text: L),

text: with

text: Fourier

text: series

text: f(x)

text: with

text: fˆ

text: f(x)e

text: l

text: l

text: l=−∞

text: Ω

text: B

text: its

text: truncated

text: series

text: is

text: deﬁned

text: as

text: the

text: projection

text: onto

text: the

text: space

text: N

text: of

text: trigonometric

text: polyno-

text: mials

text: in

text: x

text: of

text: degree

text: up

text: to

text: N,

text: given

text: by

text: P

text: N

text: f(x)

text: N

text: l

text: l=

text: N


text: ∞ ∗ ∞ ∗ only an (cid:96) (0, T ; (cid:96)2) convergence for u, and (cid:96) (0, T ; (cid:96)2) convergence can be obtained for ψ, at the theoretical level; see the detailed discussions in an earlier work [17]. In addition, a severe time ≤ step constraint, ∆t Ch2, has to be imposed to ensure a convergence in that approach, compared to the unconditional convergence established in this article. 5 Numerical Results In this section we perform a numerical accuracy check for the fully discrete pseudospectral scheme (2.11). Similar to [17], the exact solitary wave solution of the GB equation (with p = 2) is given by (cid:18) (cid:19) − P − u (x, t) = Asech2 (x c t) , (5.1) e 2 0 ≤ in which 0 < P 1. In more detail, the amplitude A, the wave speed c and the real 0 parameter P satisfy (cid:0) (cid:1) 3P2 − A = , c = 1 P2 1/2 . (5.2) 2 | | → ∞ Since the exact proﬁle (5.1) decays exponentially as x , it is natural to apply Fourier − pseudospectral approximation on an interval ( L, L), with L large enough. In this numerical − experiment, we set the computational domain as Ω = ( 40, 40). A moderate amplitude A = 0.5 is chosen in the test. 5.1 Spectral convergence in space − To investigate the accuracy in space, we ﬁx ∆t = 10 4 so that the temporal numerical error is negligible. We compute solutions with grid sizes N = 32 to N = 128 in increments of 8, and we solve up to time T = 4. The following numeric(cid:13)al errors at th(cid:13)is ﬁnal time (cid:13) (cid:13) (cid:107) − (cid:107) − ψ ψ , and D2 (u u ) , (5.3) e 2 N e 2 are presented in Fig. 1. The spatial spectral accuracy is apparently observed for both u and − ψ = u . Due to the ﬁxed time step ∆t = 10 4, a saturation of spectral accuracy appears with an t increasing N. 5.2 Second order convergence in time To explore the temporal accuracy, we ﬁx the spatial resolution as N = 512 so that the numerical error is dominated by the temporal ones. We compute solutions with a sequence of time step sizes, ∆t = T , with N = 100 to N = 1000 in increments of 100, and T = 4. Fig. 2 NK K K 22 ##LTLine## ##LTLine## ##LTLine##

text: only

text: an

text: T

text: convergence

text: for

text: u,

text: and

text: T

text: convergence

text: can

text: be

text: obtained

text: for

text: ψ,

text: at

text: the

text: theoretical

text: level;

text: see

text: the

text: detailed

text: discussions

text: in

text: an

text: earlier

text: work

text: In

text: addition,

text: a

text: severe

text: time

text: step

text: constraint,

text: ∆t

text: has

text: to

text: be

text: imposed

text: to

text: ensure

text: a

text: convergence

text: in

text: that

text: approach,

text: compared

text: to

text: the

text: unconditional

text: convergence

text: established

text: in

text: this

text: article.

text: Numerical

text: Results

text: In

text: this

text: section

text: we

text: perform

text: a

text: numerical

text: accuracy

text: check

text: for

text: the

text: fully

text: discrete

text: pseudospectral

text: scheme

text: Similar

text: to

text: the

text: exact

text: solitary

text: wave

text: solution

text: of

text: the

text: GB

text: equation

text: (with

text: p

text: is

text: given

text: by

text: P

text: u

text: (x,

text: t)

text: (x

text: c

text: t)

text: e

text: in

text: which

text: P

text: In

text: more

text: detail,

text: the

text: amplitude

text: A,

text: the

text: wave

text: speed

text: c

text: and

text: the

text: real

text: parameter

text: P

text: satisfy

text: A

text: c

text: Since

text: the

text: exact

text: proﬁle

text: decays

text: exponentially

text: as

text: x

text: it

text: is

text: natural

text: to

text: apply

text: Fourier

text: pseudospectral

text: approximation

text: on

text: an

text: interval

text: L,

text: L),

text: with

text: L

text: large

text: enough.

text: In

text: this

text: numerical

text: experiment,

text: we

text: set

text: the

text: computational

text: domain

text: as

text: Ω

text: A

text: moderate

text: amplitude

text: A

text: is

text: chosen

text: in

text: the

text: test.

text: Spectral

text: convergence

text: in

text: space

text: To

text: investigate

text: the

text: accuracy

text: in

text: space,

text: we

text: ﬁx

text: ∆t

text: so

text: that

text: the

text: temporal

text: numerical

text: error

text: is

text: negligible.

text: We

text: compute

text: solutions

text: with

text: grid

text: sizes

text: N

text: to

text: N

text: in

text: increments

text: of

text: and

text: we

text: solve

text: up

text: to

text: time

text: T

text: The

text: following

text: errors

text: at

text: ﬁnal

text: time

text: ψ

text: ψ

text: and

text: (u

text: u

text: e

text: N

text: e

text: are

text: presented

text: in

text: Fig.

text: The

text: spatial

text: spectral

text: accuracy

text: is

text: apparently

text: observed

text: for

text: both

text: u

text: and

text: ψ

text: u

text: Due

text: to

text: the

text: ﬁxed

text: time

text: step

text: ∆t

text: a

text: saturation

text: of

text: spectral

text: accuracy

text: appears

text: with

text: an

text: t

text: increasing

text: N.

text: Second

text: order

text: convergence

text: in

text: time

text: To

text: explore

text: the

text: temporal

text: accuracy,

text: we

text: ﬁx

text: the

text: spatial

text: resolution

text: as

text: N

text: so

text: that

text: the

text: numerical

text: error

text: is

text: dominated

text: by

text: the

text: temporal

text: ones.

text: We

text: compute

text: solutions

text: with

text: a

text: sequence

text: of

text: time

text: step

text: sizes,

text: ∆t

text: T

text: with

text: N

text: to

text: N

text: in

text: increments

text: of

text: and

text: T

text: Fig.

text: NK

text: K

text: K

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 1. Introduction Sentiment analysis (Pang and Lee 2008; Liu 2012) has received much attention from both research and industry communities in recent years. Sentiment classiﬁcation, which identiﬁes sentiment polarity (positive or negative) from text (sentence or document), has been the most extensively studied task in sentiment analysis. Up until now, there have been two mainstream approaches for sentiment classiﬁcation. The lexicon-based approach (Turney 2002; Taboada et al. 2011) aims to aggregate the sentiment polarity of a sentence from the polarity of words or phrases found in the sentence, while the learning-based approach (Pang, Lee, and Vaithyanathan 2002) treats sentiment polarity identiﬁcation as a special text classiﬁcation task and focuses on building classiﬁers from a set of sentences (or documents) annotated with their corresponding sentiment polarity. The lexicon-based sentiment classiﬁcation approach is simple and interpretable, but suffers from scalability and is inevitably limited by sentiment lexicons that are commonly created manually by experts. It has been widely recognized that sentiment expressions are colloquial and evolve over time very frequently. Taking tweets from Twitter1 and movie reviews in IMDB2 as examples, people use very casual language as well as informal and new vocabulary to comment on general topics and movies. In practice, it is not feasible to create and maintain sentiment lexicons to capture sentiment expressions with high coverage. On the other hand, the learning-based approach relies on large annotated samples to overcome the vocabulary coverage and deals with varia- tions of words in sentences. Human ratings in reviews (Maas et al. 2011) and emoticons in tweets (Davidov, Tsur, and Rappoport 2010; Zhao et al. 2012) are extensively used to collect a large number of training corpora to train the sentiment classiﬁer. However, it is usually not easy to design effective features to build the classiﬁer. Among others, un- igrams have been reported as the most effective features (Pang, Lee, and Vaithyanathan 2002) in sentiment classiﬁcation. Handling complicated expressions delivering people’s opinions is one of the most challenging problems in sentiment analysis. Among others, compositionalities such as negation, intensiﬁcation, contrast, and their combinations are typical cases. We show some concrete examples below. (1) The movie is not good. [negation] (2) The movie is very good. [intensiﬁcation] (3) The movie is not funny at all. [negation + intensiﬁcation] (4) The movie is just so so, but i still like it. [contrast] (5) The movie is not very good, but i still like it. [negation + intensiﬁcation + contrast] The negation expressions, intensiﬁcation modiﬁers, and the contrastive conjunction can change the polarity ((1), (3), (4), (5)), strength ((2), (3), (5)), or both ((3), (5)) of the sentiment of the sentences. We do not need any detailed explanations here as 1 http://twitter.com 2 http://www.imdb.com 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: UKDWJN+URWPalladioL-Bold

text: Introduction

text: Sentiment

font: KOHCYW+URWPalladioL-Roma

text: analysis

text: (Pang

text: and

text: Lee

text: Liu

text: has

text: received

text: much

text: attention

text: from

text: both

text: research

text: and

text: industry

text: communities

text: in

text: recent

text: years.

text: Sentiment

text: classiﬁcation,

text: which

text: identiﬁes

text: sentiment

text: polarity

text: (positive

text: or

text: negative)

text: from

text: text

text: (sentence

text: or

text: document),

text: has

text: been

text: the

text: most

text: extensively

text: studied

text: task

text: in

text: sentiment

text: analysis.

text: Up

text: until

text: now,

text: there

text: have

text: been

text: two

text: mainstream

text: approaches

text: for

text: sentiment

text: classiﬁcation.

text: The

text: lexicon-based

text: approach

text: (Turney

text: Taboada

text: et

text: al.

text: aims

text: to

text: aggregate

text: the

text: sentiment

text: polarity

text: of

text: a

text: sentence

text: from

text: the

text: polarity

text: of

text: words

text: or

text: phrases

text: found

text: in

text: the

text: sentence,

text: while

text: the

text: learning-based

text: approach

text: (Pang,

text: Lee,

text: and

text: Vaithyanathan

text: treats

text: sentiment

text: polarity

text: identiﬁcation

text: as

text: a

text: special

text: text

text: classiﬁcation

text: task

text: and

text: focuses

text: on

text: building

text: classiﬁers

text: from

text: a

text: set

text: of

text: sentences

text: (or

text: documents)

text: annotated

text: with

text: their

text: corresponding

text: sentiment

text: polarity.

text: The

text: lexicon-based

text: sentiment

text: classiﬁcation

text: approach

text: is

text: simple

text: and

text: interpretable,

text: but

text: suffers

text: from

text: scalability

text: and

text: is

text: inevitably

text: limited

text: by

text: sentiment

text: lexicons

text: that

text: are

text: commonly

text: created

text: manually

text: by

text: experts.

text: It

text: has

text: been

text: widely

text: recognized

text: that

text: sentiment

text: expressions

text: are

text: colloquial

text: and

text: evolve

text: over

text: time

text: very

text: frequently.

text: Taking

text: tweets

text: from

text: and

text: movie

text: reviews

text: in

text: as

text: examples,

text: people

text: use

text: very

text: casual

text: language

text: as

text: well

text: as

text: informal

text: and

text: new

text: vocabulary

text: to

text: comment

text: on

text: general

text: topics

text: and

text: movies.

text: In

text: practice,

text: it

text: is

text: not

text: feasible

text: to

text: create

text: and

text: maintain

text: sentiment

text: lexicons

text: to

text: capture

text: sentiment

text: expressions

text: with

text: high

text: coverage.

text: On

text: the

text: other

text: hand,

text: the

text: learning-based

text: approach

text: relies

text: on

text: large

text: annotated

text: samples

text: to

text: overcome

text: the

text: vocabulary

text: coverage

text: and

text: deals

text: with

text: varia-

text: tions

text: of

text: words

text: in

text: sentences.

text: Human

text: ratings

text: in

text: reviews

text: (Maas

text: et

text: al.

text: and

text: emoticons

text: in

text: tweets

text: (Davidov,

text: Tsur,

text: and

text: Rappoport

text: Zhao

text: et

text: al.

text: are

text: extensively

text: used

text: to

text: collect

text: a

text: large

text: number

text: of

text: training

text: corpora

text: to

text: train

text: the

text: sentiment

text: classiﬁer.

text: However,

text: it

text: is

text: usually

text: not

text: easy

text: to

text: design

text: effective

text: features

text: to

text: build

text: the

text: classiﬁer.

text: Among

text: others,

text: un-

text: igrams

text: have

text: been

text: reported

text: as

text: the

text: most

text: effective

text: features

text: (Pang,

text: Lee,

text: and

text: Vaithyanathan

text: in

text: sentiment

text: classiﬁcation.

text: Handling

text: complicated

text: expressions

text: delivering

text: people’s

text: opinions

text: is

text: one

text: of

text: the

text: most

text: challenging

text: problems

text: in

text: sentiment

text: analysis.

text: Among

text: others,

text: compositionalities

text: such

text: as

text: negation,

text: intensiﬁcation,

text: contrast,

text: and

text: their

text: combinations

text: are

text: typical

text: cases.

text: We

text: show

text: some

text: concrete

text: examples

text: below.

text: The

text: movie

text: is

text: not

text: good.

text: [negation]

text: The

text: movie

text: is

text: very

text: good.

text: [intensiﬁcation]

text: The

text: movie

text: is

text: not

text: funny

text: at

text: all.

text: [negation

text: intensiﬁcation]

text: The

text: movie

text: is

text: just

text: so

text: so,

text: but

text: i

text: still

text: like

text: it.

text: [contrast]

text: The

text: movie

text: is

text: not

text: very

text: good,

text: but

text: i

text: still

text: like

text: it.

text: [negation

text: intensiﬁcation

text: contrast]

text: The

text: negation

text: expressions,

text: intensiﬁcation

text: modiﬁers,

text: and

text: the

text: contrastive

text: conjunction

text: can

text: change

text: the

text: polarity

text: strength

text: or

text: both

text: of

text: the

text: sentiment

text: of

text: the

text: sentences.

text: We

text: do

text: not

text: need

text: any

text: detailed

text: explanations

text: here

text: as

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: → (X wj) i [i, X, j] → (X wi1X wj ) [i , X , j ] i 1 j1 1 1 1 [i, X, j] → (X wi1X wi2X wj ) [i , X , j ] [i , X , j ] i 1 j1 2 j2 1 1 1 2 2 2 [i, X, j] → (X X X ) [i, X , k] [k, X , j] 1 2 1 2 [i, X, j] E → ( wj) E i [i, , j] → E E (X X ) [i, , k] [k, X , j] 1 1 [i, X, j] → E E (X X ) [i, X , k] [k, , j] 1 1 [i, X, j] where X, X , X represent N or P. 1 2 Figure 3 Inference rules for the basic parsing model. (cid:88) { } A(ψ; s, T) = log exp φ(s, t)Tψ (5) ∈ t T(s) where A(ψ; s, T) is the log-partition function with respect to T(s). The log-linear model is a discriminative model, and it is widely used in natural language processing. We can use φ(s, t)Tψ as the score of the parse tree without normalization in the decoding | ∝ process, because p(t s; T, ψ) φ(s, t)Tψ and this will not change the ranking order. 3.4 Polarity Model The goal of the polarity model is to model the calculation of sentiment strength and polarity of a text span from its sub-spans in the parsing process. It is speciﬁed in terms of the rules employed in the parsing process. We expand the notations in the inference rule (3) to incorporate the polarity model. The new form of inference rule is: (r) H Φ . . . H Φ 1 1 K K (6) [i, X, j]Φ 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (X

text: wj)

text: i

text: [i,

text: X,

text: j]

text: (X

text: wj

text: [i

text: X

text: j

text: i

text: [i,

text: X,

text: j]

text: (X

text: wj

text: [i

text: X

text: j

text: [i

text: X

text: j

text: i

text: [i,

text: X,

text: j]

text: (X

text: X

text: X

text: [i,

text: X

text: k]

text: [k,

text: X

text: j]

text: [i,

text: X,

text: j]

text: E

text: wj)

text: E

text: i

text: [i,

text: j]

text: E

text: E

text: (X

text: X

text: [i,

text: k]

text: [k,

text: X

text: j]

text: [i,

text: X,

text: j]

text: E

text: E

text: (X

text: X

text: [i,

text: X

text: k]

text: [k,

text: j]

text: [i,

text: X,

text: j]

text: where

text: X,

text: X

text: X

text: represent

text: N

text: or

text: P.

text: Figure

text: Inference

text: rules

text: for

text: the

text: basic

text: parsing

text: model.

text: A(ψ;

text: s,

text: T)

text: log

text: exp

text: φ(s,

text: t)Tψ

text: t

text: T(s)

text: where

text: A(ψ;

text: s,

text: T)

text: is

text: the

text: log-partition

text: function

text: with

text: respect

text: to

text: T(s).

text: The

text: log-linear

text: model

text: is

text: a

text: discriminative

text: model,

text: and

text: it

text: is

text: widely

text: used

text: in

text: natural

text: language

text: processing.

text: We

text: can

text: use

text: φ(s,

text: t)Tψ

text: as

text: the

text: score

text: of

text: the

text: parse

text: tree

text: without

text: normalization

text: in

text: the

text: decoding

text: process,

text: because

text: p(t

text: s;

text: T,

text: ψ)

text: φ(s,

text: t)Tψ

text: and

text: this

text: will

text: not

text: change

text: the

text: ranking

text: order.

text: Polarity

text: Model

text: The

text: goal

text: of

text: the

text: polarity

text: model

text: is

text: to

text: model

text: the

text: calculation

text: of

text: sentiment

text: strength

text: and

text: polarity

text: of

text: a

text: text

text: span

text: from

text: its

text: sub-spans

text: in

text: the

text: parsing

text: process.

text: It

text: is

text: speciﬁed

text: in

text: terms

text: of

text: the

text: rules

text: employed

text: in

text: the

text: parsing

text: process.

text: We

text: expand

text: the

text: notations

text: in

text: the

text: inference

text: rule

text: to

text: incorporate

text: the

text: polarity

text: model.

text: The

text: new

text: form

text: of

text: inference

text: rule

text: is:

text: (r)

text: H

text: Φ

text: H

text: Φ

text: K

text: K

text: [i,

text: X,

text: j]Φ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: h(x) 1.0 0.5 Linear Function Logistic Function 0.0 x Figure 5 Logistic function h(x) = 1{− } truncates polarity values to (0,1) smoothly. The computed 1+exp x values are used as polarity probabilities. (cid:40) → (N P but N) [i , P, j ]Φ [i , N, j ]Φ 1 1 1 2 2 2 (12) N| P| N| P( wj) = h(θ + θ P( wj1) + θ P( wj2)) [i, N, j] P| i −0 N1| i1 2 i2 P( wj) = 1 P( wj) i i → N| where the polarity probability of the rule N P but N is computed by P( wj) = P| N| i h(θ + θ P( wj1) + θ P( wj2)). It can express the contrast relation by speciﬁc pa- ram0eters1 θ , θ , ia1nd θ .2 i2 0 1 2 It should be noted that a linear regression model could turn out to be problem- atic, as it may produce unreasonable results. For example, if we do not add any con- N| − P| P| straint, we may get P( wj) = 0.6 + P( wj ). When P( wj ) = 0.55, we will N| − i − i+1 i+1 get P( wj) = 0.6 + 0.55 = 0.05. It conﬂicts with the deﬁnition that the polarity i probability ranges from zero to one. Figure 5 intuitively shows that the logistic function truncates polarity values to (0, 1) smoothly. 3.5 Constraints We incorporate additional constraints into the parsing model. They are used as pruning conditions in the derivation process not only to improve efﬁciency but also to force the derivation towards the correct direction. We expand the inference rules in Section as, (r) H Φ . . . H Φ 1 1 K K C (13) [i, X, j]Φ where C is a side condition. The constraints are interpreted in a Boolean manner. If the constraint C is satisﬁed, the rule can be used, otherwise, it cannot. We deﬁne two constraints in the parsing model. 16 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: h(x)

text: Linear

text: Function

text: Logistic

text: Function

text: x

text: Figure

text: Logistic

text: function

text: h(x)

text: truncates

text: polarity

text: values

text: to

text: smoothly.

text: The

text: computed

text: x

text: values

text: are

text: used

text: as

text: polarity

text: probabilities.

text: (N

text: P

text: but

text: N)

text: [i

text: P,

text: j

text: ]Φ

text: [i

text: N,

text: j

text: ]Φ

text: N|

text: P|

text: N|

text: P(

text: wj)

text: h(θ

text: θ

text: P(

text: θ

text: P(

text: [i,

text: N,

text: j]

text: P|

text: i

text: −0

text: P(

text: wj)

text: P(

text: wj)

text: i

text: i

text: N|

text: where

text: the

text: polarity

text: probability

text: of

text: the

text: rule

text: N

text: P

text: but

text: N

text: is

text: computed

text: by

text: P(

text: wj)

text: P|

text: N|

text: i

text: h(θ

text: θ

text: P(

text: θ

text: P(

text: It

text: can

text: express

text: the

text: contrast

text: relation

text: by

text: speciﬁc

text: pa-

text: θ

text: θ

text: θ

text: It

text: should

text: be

text: noted

text: that

text: a

text: linear

text: regression

text: model

text: could

text: turn

text: out

text: to

text: be

text: problem-

text: atic,

text: as

text: it

text: may

text: produce

text: unreasonable

text: results.

text: For

text: example,

text: if

text: we

text: do

text: not

text: add

text: any

text: con-

text: N|

text: P|

text: P|

text: straint,

text: we

text: may

text: get

text: P(

text: wj)

text: P(

text: wj

text: When

text: P(

text: wj

text: we

text: will

text: N|

text: i

text: get

text: P(

text: wj)

text: It

text: conﬂicts

text: with

text: the

text: deﬁnition

text: that

text: the

text: polarity

text: i

text: probability

text: ranges

text: from

text: zero

text: to

text: one.

text: Figure

text: intuitively

text: shows

text: that

text: the

text: logistic

text: function

text: truncates

text: polarity

text: values

text: to

text: smoothly.

text: Constraints

text: We

text: incorporate

text: additional

text: constraints

text: into

text: the

text: parsing

text: model.

text: They

text: are

text: used

text: as

text: pruning

text: conditions

text: in

text: the

text: derivation

text: process

text: not

text: only

text: to

text: improve

text: efﬁciency

text: but

text: also

text: to

text: force

text: the

text: derivation

text: towards

text: the

text: correct

text: direction.

text: We

text: expand

text: the

text: inference

text: rules

text: in

text: Section

text: as,

text: (r)

text: H

text: Φ

text: H

text: Φ

text: K

text: K

text: C

text: [i,

text: X,

text: j]Φ

text: where

text: C

text: is

text: a

text: side

text: condition.

text: The

text: constraints

text: are

text: interpreted

text: in

text: a

text: Boolean

text: manner.

text: If

text: the

text: constraint

text: C

text: is

text: satisﬁed,

text: the

text: rule

text: can

text: be

text: used,

text: otherwise,

text: it

text: cannot.

text: We

text: deﬁne

text: two

text: constraints

text: in

text: the

text: parsing

text: model.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##