Research Article Symplectic Schemes for Linear Stochastic ...downloads.hindawi.com/journals/aaa/2014/427023.pdf · Symplectic Schemes for Linear Stochastic Schrödinger Equations

Research ArticleSymplectic Schemes for Linear Stochastic SchroumldingerEquations with Variable Coefficients

Xiuling Yin12 and Yanqin Liu12

1 School of Mathematical Sciences Dezhou University Dezhou 253023 China2The Center of Data Processing and Analyzing Dezhou University Dezhou 253023 China

Correspondence should be addressed to Xiuling Yin yinxiulingdzueducn

Received 16 December 2013 Accepted 4 April 2014 Published 22 April 2014

Academic Editor Adem Kılıcman

Copyright copy 2014 X Yin and Y Liu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a kind of symplectic schemes for linear Schrodinger equations with variable coefficients and a stochasticperturbation term by using compact schemes in space The numerical stability property of the schemes is analyzed The schemespreserve a discrete charge conservation law They also follow a discrete energy transforming formula The numerical experimentsverify our analysis

1 Introduction

Differential equations (DEs) are important models in sci-ences and engineering By theoretical and numerical analysisof DEs we can yield some mathematical explanation ofmany phenomena in applied sciences [1ndash5] Time-dependentSchrodinger equations arise in quantum physics opticsand many other fields [6 7] Some numerical methodsfor such equations such as symplectic scheme and mul-tisymplectic schemes have been proposed in [8ndash14] Theschemes possess good numerical stability Compact schemesare popular recently due to high accuracy compactnessand economic resource in scientific computation [15ndash17]In this paper applying compact operators we constructsymplectic methods to the initial boundary problems of thelinear Schrodinger equation with a variable coefficient and astochastic perturbation term (denoted by LSES)

119894119906119905+ 1199061199094 + 119891 (119909) 119906 = 120598119906 ∘ 120594 119909 isin [0 119871]

119906 (119909 0) = 1199060(119909) 119905 isin [0 119879]

119906 (0 119905) = 119906 (119871 119905)

(1)

where 1198942 = minus1 119891(119909) is a real differential function 1199060(119909) is a

differential function 120598 is a small real number and ∘ means

Stratonovich product 120594 is a real-valued white noise which isdelta correlated in time and either smooth or delta correlatedin space For an integer 119898 119906

119909119898 and 119906

119905119898 mean the 119898-order

partial derivatives of 119906 with respect to 119909 and 119905 respectivelyThe system (1) with 120598 = 0 is a deterministic system When 120598

is small we can think that (1) is perturbed by the stochasticterm

By multiplying (1) by 119906 or 119906119905and then integrating it with

respect to 119905 and 119909 it is easy to verify the following result

Proposition 1 Under the periodic boundary condition

(a) the solution of (1) satisfies the charge conservation law

Q (119905) = int

119871

0

|119906 (119909 119905)|2d119909 = Q (0) (2)

(b) the corresponding deterministic system (120598 = 0) of (1)possesses the energy conservation law

E (119905) = int

119871

0

10038161003816100381610038161199061199091199091003816100381610038161003816

2

+ 119891 (119909) |119906|2d119909 = E (0) (3)

The paper is organized as follows In Section 2 we give asymplectic structure of the LSES In Section 3 we present the

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 427023 7 pageshttpdxdoiorg1011552014427023

2 Abstract and Applied Analysis

new symplectic methods to the LSES First we use a kind ofcompact schemes in discretization of spatial derivativeThenin temporal discretization we adopt the symplectic midpointmethod The new methods are denoted by LSC schemes Wealso analyze the numerical stability of LSC schemes We givetwo numerical examples to support our theory in Section 4At last we make some conclusions

2 Symplectic Structure of the LSES

Let 119906 = 119901 + 119894119902 The LSES (1) can be written in

119901119905+ 1199021199094 + 119891 (119909) 119902 = 120598119902 ∘ 120594

minus119902119905+ 1199011199094 + 119891 (119909) 119901 = 120598119901 ∘ 120594

(4)

Introducing the variable 119911 = (119901 119902)119879 (4) reads in stochas-

tic symplectic context

119911119905= 119869minus1nabla119911119867(119911) + 120598119869

minus1nabla119911119878 (119911) ∘ 120594 (5)

where

119867(119911) =1

2(1199012

119909119909+ 1199022

119909119909) +

119891 (119909)

2(1199012+ 1199022)

119878 (119911) = minus1

2(1199012+ 1199022)

119869 = (0 1

minus1 0)

(6)

The system satisfies the symplectic conservation law [7 1218]

120596119905= 0 120596 = 119889119901 and 119889119902 (7)

Numerical methods which preserve the discrete symplecticconservation law are called symplectic methods Symplecticmethods have good numerical stability

3 LSC Schemes

31 Compact Scheme Introduce the following uniformmeshgrids

119909119896= 119896ℎ 119896 = 0 1 119873 119905

119899= 119899120591 119899 = 0 1

(8)

where ℎ = 119871119873 and 120591 are spatial and temporal step sizesrespectively Denote the numerical values of 119906(119909

119896 119905119899) at the

nodes (119909119896 119905119899) by 119906

119899

119896 The symbols 119906

119899 and 119906119896mean the

numerical solution vectors at 119905 = 119905119899and 119909 = 119909

119896with compo-

nents 119906119899119896 respectively Furthermore we denote

119906119899+(12)

119896=

119906119899+1

119896+ 119906119899

119896

2 120575

119905119906119899+(12)

119896=

119906119899+1

119896minus 119906119899

119896

120591 (9)

Introducing the following linear operators

A119906119896= 120572119906119896minus1

+ 119906119896+ 120572119906119896+1

B119906119896= 119887

119906119896+3

minus 9119906119896+1

+ 16119906119896minus 9119906119896minus1

+ 119906119896minus3

6ℎ4

+ 119886119906119896+2

minus 4119906119896+1

+ 6119906119896minus 4119906119896minus1

+ 119906119896minus2

ℎ4

(10)

we adopt formula [19]

1205754

119909119906119896= Aminus1B119906119896

(11)

to approximate 1199061199094 which means that

A1205754

119909119906119896= B119906

119896 (12)

By Taylor expansion we can derive a family of fourth-orderschemes with

119886 = 2 (1 minus 120572) 119887 = 4120572 minus 1 (13)

The leading termof the truncation error of themethod is ((7minus26120572)240)(119906

1199098)119896ℎ4 If 119887 = 0 we get a scheme with smaller

stencil A sixth-order scheme is obtained with

120572 =7

26 119886 =

19

13 119887 =

1

13 (14)

Denote two symmetric and cyclic matrices by

119860 =

[[[[[[[[[[[[[[[[

[

1 120572 0 sdot sdot sdot sdot sdot sdot 0 120572

120572 1 120572 0 sdot sdot sdot sdot sdot sdot 0

0 120572 1 120572 0 sdot sdot sdot 0

d d d d d

d 120572 1 120572 0

0 0 sdot sdot sdot 0 120572 1 120572

120572 0 sdot sdot sdot sdot sdot sdot 0 120572 1

]]]]]]]]]]]]]]]]

]119873times119873

Abstract and Applied Analysis 3

119861 =1

6ℎ4

[[[[[[[[[[[[[[[[[[

[

1198880

1198881

6119886 119887 0 sdot sdot sdot sdot sdot sdot 0 119887 6119886 1198881

1198881

1198880

1198881

6119886 119887 0 sdot sdot sdot sdot sdot sdot 0 119887 6119886

6119886 1198881

1198880

1198881


119887 6119886 1198881

1198880

1198881

6119886 119887 0 sdot sdot sdot sdot sdot sdot 0

0 119887 6119886 1198881

1198880

1198881

6119886 119887 0 sdot sdot sdot 0

d d d d d d d d d

0 sdot sdot sdot 0 119887 6119886 1198881

1198880

1198881

6119886 119887 0

0 sdot sdot sdot sdot sdot sdot 0 119887 6119886 1198881

1198880

1198881

6119886 119887

119887 0 sdot sdot sdot sdot sdot sdot 0 119887 6119886 1198881

1198880

1198881

6119886


1198880

1198881

1198881


1198880

]]]]]]]]]]]]]]]]]]

]119873times119873

(15)

where 1198880= 16119887 + 36119886 and 119888

1= minus9119887 minus 24119886 Then the matrix

form of (11) is

1205754

119909119906119899= 119860minus1119861119906119899 (16)

32 Discretization of the LSES Applying the approximation(11) to linear system (4) we obtain the following semidis-cretization stochastic Hamiltonian system

(119911119896)119905= 119869minus1nabla119911119867(119911119896) + 120598119869

minus1nabla119911119878 (119911119896) ∘ 120594119896 (17)

where

119867(119911) =119891 (119909119896)

2(1199012

119896+ 1199022

119896) +

1

21205754

119909(1199012

119896+ 1199022

119896)

119878 (119911) = minus1

2(1199012

119896+ 1199022

119896)

(18)

In temporal discretization of (17) we apply the symplecticmidpoint method

120575119905119911119899+(12)

119896= 119869minus1nabla119911119867(119911119899+(12)

119896) + 120598119869

minus1nabla119911119867(119911119899+(12)

119896)

∘ 120594119899+(12)

119896

(19)

Its componentwise formulation is

119901119899+1

119896minus 119901119899

119896

120591= minus119891 (119909

119896) 119902119899+(12)

119896minus 1205754

119909119902119899+(12)

119896

+ 120598119902119899+(12)

119896∘ 120594119899+(12)

119896

119902119899+1

119896minus 119902119899

119896

120591= 119891 (119909

119896) 119901119899+(12)

119896+ 1205754

119909119901119899+(12)

119896

minus 120598119901119899+(12)

119896∘ 120594119899+(12)

119896

(20)

According to the Fourier analysis the LSC schemes (19)are unconditionally stable In fact we can derive

1205754

119909119911119899

119896= 120583119911119899

119896 (21)

where

120583 =4(120596 minus 1)

2(3119886 + 2119887 + 119887120596)

3ℎ4 (1 + 2120572120596) 120596 = cos120573ℎ (22)

Then with (19) and (21) we can obtain 119911119899+1

= 119866119911119899 with

(1 +11988821205912

4)119866 = (

1 minus11988821205912

4minus119888120591

119888120591 1 minus11988821205912

4

) (23)

where 119888 = 120583 + 119891(119909119896) minus 120598 ∘ 120594

119899+(12)

119896 By direct computation

we can derive that the spectral radius of the matrix 119866 is 1and 119866

2= 1 Therefore the scheme (19) is unconditionally

stable Moreover by symmetry they are nondissipative

Theorem 2 Let 1199111198992 = ℎsum119896119911119899

119896119911119899

119896 Then 119911119899 is the discrete

charge invariant of the LSC schemes (19) which implies thediscrete charge conservation law of (2)

Scheme (19) can be rewritten in compact form

119894A119906119899+1

119896minus 119906119899

119896

120591+B119906

119899+(12)

119896+A119891 (119909

119896) 119906119899+(12)

119896

= 120598A119906119899+(12)

119896∘ 120594119899+(12)

119896

(24)

Multiplying (24) by (119906119899+1119896

minus119906119899

119896) and summing over 119896 we obtain

119894120591

2sum

119896

[B119906119899+1

119896119906119899+1

119896minusB119906

119899

119896119906119899

119896]

+119894120591

2sum

119896

A119891 (119909119896) [119906119899+1

119896119906119899+1

119896minus 119906119899

119896119906119899

119896]

minus119894120591120598

2sum

119896

[A119906119899+1

119896∘ 120594119899+(12)

119896119906119899+1

119896minusA119906119899

119896∘ 120594119899+(12)

119896119906119899

119896]

+119894120591

2sum

119896

[B119906119899

119896119906119899+1

119896minusB119906

119899+1

119896119906119899

119896]

+119894120591

2sum

119896

A119891 (119909119896) [119906119899

119896119906119899+1

119896minus 119906119899+1

119896119906119899

119896]

minus119894120591120598

2sum

119896

[A119906119899

119896∘ 120594119899+(12)

119896119906119899+1

119896minusA119906119899+1

119896∘ 120594119899+(12)

119896119906119899

119896]

= sum

119896

[A (119906119899+1

119896minus 119906119899

119896) (119906119899+1

119896minus 119906119899

119896)]

(25)


0

5

10

0

2

4

0

05

1

15

tx

|un k|

(a) 120598 = 005

0

5

10

0

2

4

0

05

1

15

tx

|un k|

(b) 120598 = 01

Figure 1 |119906119899119896| for one trajectory

0 10 20 30 40098

1

102

104

106

108

t

||u||

(a)

0 2 4 6 8 100

05

1

15

2

times10minus12

t

|en Q|

(b)

Figure 2 119906infinfor one trajectory and the average norm over 50 trajectories (a) Residuals of discrete charge for one trajectory (b)

Since A and B are symmetric the first three summationterms in the above equality are purely imaginary while thelast four summation terms are real Denote

119864119899= ℎsum

119896

[B119906119899

119896119906119899

119896] + ℎsum

119896

[A119891 (119909119896) 119906119899

119896119906119899

119896]

minusℎ

2120598sum

119896

[A119906119899

119896∘ 120594119899

119896119906119899

119896]

(26)

Now taking the imaginary parts of (25) we can get that

119864119899+1

minus 119864119899=ℎ

2120598sum

119896

[A119906119899+1

119896∘ 120594119899

119896119906119899+1

119896minusA119906119899

119896∘ 120594119899+1

119896119906119899

119896] (27)

Denote

V119899119896= radic

119887

6

119906119899

119896+3minus 119906119899

119896

ℎ2 V119899

119896= radic

3119887

2

119906119899

119896+1minus 119906119899

119896

ℎ2

119908119899

119896= radic119886

119906119899

119896+2minus 119906119899

119896

ℎ2

119908119899

119896= 2radic119886

119906119899

119896+1minus 119906119899

119896

ℎ2 119910

119899

119896= A119891 (119909

119896) 119906119899

119896

119910119899

119896= A119906

119899

119896∘ 120594119899

119896

⟨119906119899 119910119899⟩ = ℎsum

119896

119906119899

119896119910119899

119896

10038171003817100381710038171199061198991003817100381710038171003817

2

= ⟨119906119899 119906119899⟩

(28)


0 5 10 15 200

001

002

003

004

005

006

007

008

t

|en H|

(a) 119890119899119867

0 5 10 15 200

02

04

06

08

1

t

HnminusH

0

(b) 119867119899 minus 1198670

Figure 3 Residuals of discrete energy for one trajectory and the average energy over 50 trajectories

0

20

40

0

2

4

0

05

1

15

tx

|un k|

(a) 120598 = 005

0

20

40

0

2

4

0

05

1

15

tx

|un k|

(b) 120598 = 01


According to the Green formula we obtain that

ℎsum

119896

[B119906119899

119896119906119899

119896] =

1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

(29)

Then

119864119899=1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

+ ⟨119910119899 119906119899⟩ minus

120598

2⟨119910119899 119906119899⟩

(30)

Therefore from the above analysis we give the followingresult

Theorem 3 Under the periodic boundary condition the LSCschemes (19) satisfy the discrete energy transforming law (27)

4 Numerical Examples

We use the LSC scheme to solve the LSESs and investigate itsnumerical behavior According to the precise mathematicaldefinition of thewhite noise [13 14] we can simulate the noiseas 120594119899+(12)

119896= (1radicℎ120591)120594

119899+(12)

119896 where 120594119899+(12)

119896 119896 = 0 1 119873

is a sequence of independent random variables with normallawN(0 1) at each time increment Denote

10038171003817100381710038171199061198991003817100381710038171003817infin = max

1le119896le119873

1003816100381610038161003816119906119899

119896

1003816100381610038161003816 119890119899

119876= 119876119899minus 1198760

119890119899

119867= 119867119899minus 119867119899minus1

(31)

The numerical residuals of Q(119905) and H(119905) are measured by119890119899

119876and 119890119899

119867 respectively For numerical computation we take

120591 = 001 ℎ = 12058750 and 120598 = 005 01


0 10 20 30 40095

1

105

11

t

||u||

(a)

0 10 20 30 400

02

04

06

08

1

12

times10minus11

t

|en Q|

(b)


0 10 20 30 4001

015

02

025

03

035

04

045

05

t

|en H|

(a) 119890119899119867

0 10 20 30 400

05

1

15

2

t

HnminusH

0

(b) 119867119899 minus 1198670


Example 1 LSE with constant coefficients and periodicboundary condition

Consider119894119906119905+ 1199061199094 minus 15119906 = 120598119906 ∘ 120594 119909 isin [0 120587] 119905 gt 0

119906 (119909 0) = exp(1198941205873) cos 2119909

(32)

The exact solution of its deterministic system is 119906(119909 119905) =

exp[119894(119905 + (1205873))] cos 2119909 The right side in the above systemcan be seen as a stochastic perturbation term

Figure 1 plots the amplitude |119906119899

119896| for one trajectory

Figure 2 shows the evolution of 119906infin

for one trajectoryand the average norm over 50 trajectories We see that thewhite noise produces stochastic perturbation on the solitarywave and the size of perturbation depends on the size ofnoise Figure 2 plots the residuals of discrete charge of one

trajectory which verifies the conservation of discrete chargeof the compact schemes Figure 3 plots the residuals ofdiscrete energy for one trajectory and the average norm over50 trajectories The figure tells us that the stochastic noisemakes residuals of discrete energy increase linearly over time

Example 2 LSE with a variable coefficient and periodicboundary condition

Consider

119894119906119905+ 1199061199094 + (8 lowast cot2119909 minus 7) 119906 = 120598119906 ∘ 120594 119909 isin (0 120587) 119905 gt 0

119906 (119909 0) = sin2119909(33)

The exact solution of its deterministic system is 119906(119909 119910 119905) =

119890119894119905sin2119909





for one trajectoryand the average norm over 50 trajectories We see that thewhite noise produces stochastic perturbation on the solitarywave and the size of perturbation depends on the size ofnoise Figure 5 plots the residuals of discrete charge of onetrajectory which verifies the conservation of discrete chargeof the compact schemes Figure 6 plots the residuals ofdiscrete energy for one trajectory and the average norm over50 trajectories The figure tells us that the stochastic noisemakes residuals of discrete energy increase linearly over time

5 Conclusion

In this paper we apply a symplectic scheme in time anda kind of compact difference schemes in space to solvethe LSES The methods are unconditionally stable Underperiodic boundary conditions they preserve a discrete chargeinvariant and satisfy a discrete energy transforming law

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Xiuling Yin is supported by the Director Innovation Founda-tion of ICMSEC and AMSS the Foundation of CAS NNSFC(nos 91130003 and 11021101) Yanqin Liu is supported bythe Natural Science Foundation of Shandong Province (nosZR2013AQ005 and BS2013HZ026)

References

[1] A Ashyralyev A Hanalyev and P E Sobolevskii ldquoCoer-cive solvability of the nonlocal boundary value problem forparabolic differential equationsrdquo Abstract and Applied Analysisvol 6 no 1 pp 53ndash61 2001

[2] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourth kindsrdquo Abstract and Applied Analysis vol2013 Article ID 542839 9 pages 2013

[4] K Ivaz A Khastan and J J Nieto ldquoA numerical methodfor fuzzy differential equations and hybrid fuzzy differentialequationsrdquo Abstract and Applied Analysis vol 2013 Article ID735128 10 pages 2013

[5] C Ashyralyyev A Dural and Y Sozen ldquoFinite differencemethod for the reverse parabolic problemrdquoAbstract andAppliedAnalysis vol 2012 Article ID 294154 17 pages 2012

[6] A Phillips Introduction to Quantum Mechanics John Wiley ampSons Chichester UK 2003

[7] G Milstein and M Tretyakov Stochastic Numerics for Mathe-matical Physics Kluwer Academic Publishers Norwell MassUSA 1995

[8] J Hong and M-Z Qin ldquoMultisymplecticity of the centredbox discretization for Hamiltonian PDEs with 119898 ge 2 spacedimensionsrdquo Applied Mathematics Letters vol 15 no 8 pp1005ndash1011 2002

[9] L Kong J Hong L Wang and F Fu ldquoSymplectic integratorfor nonlinear high order Schrodinger equation with a trappedtermrdquo Journal of Computational and Applied Mathematics vol231 no 2 pp 664ndash679 2009

[10] J Hong Y Liu H Munthe-Kaas and A Zanna ldquoGlobally con-servative properties and error estimation of a multi-symplecticscheme for Schrodinger equations with variable coefficientsrdquoApplied Numerical Mathematics vol 56 no 6 pp 814ndash8432006

[11] J Hong X Liu and C Li ldquoMulti-symplectic Runge-Kutta-Nystrom methods for nonlinear Schrodinger equations withvariable coefficientsrdquo Journal of Computational Physics vol 226no 2 pp 1968ndash1984 2007

[12] Y-F Tang L Vazquez F Zhang and V M Perez-GarcıaldquoSymplectic methods for the nonlinear Schrodinger equationrdquoComputers amp Mathematics with Applications vol 32 no 5 pp73ndash83 1996

[13] S Jiang L Wang and J Hong ldquoStochastic multi-symplecticintegrator for stochastic nonlinear Schrodinger equationrdquoCom-munications in Computational Physics vol 14 no 2 pp 393ndash4112013

[14] A Debussche and L di Menza ldquoNumerical simulation offocusing stochastic nonlinear Schrodinger equationsrdquo PhysicaD vol 162 no 3-4 pp 131ndash154 2002

[15] Y Ma L Kong J Hong and Y Cao ldquoHigh-order compactsplitting multisymplectic method for the coupled nonlinearSchrodinger equationsrdquo Computers amp Mathematics with Appli-cations vol 61 no 2 pp 319ndash333 2011

[16] T V S Sekhar and B H S Raju ldquoAn efficient higher ordercompact scheme to capture heat transfer solutions in sphericalgeometryrdquo Computer Physics Communications vol 183 no 11pp 2337ndash2345 2012

[17] A Mohebbi and Z Asgari ldquoEfficient numerical algorithmsfor the solution of ldquogoodrdquo Boussinesq equation in water wavepropagationrdquo Computer Physics Communications vol 182 no12 pp 2464ndash2470 2011

[18] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in ComputationalMathematicsSpringer Berlin Germany 2nd edition 2006

[19] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


new symplectic methods to the LSES First we use a kind ofcompact schemes in discretization of spatial derivativeThenin temporal discretization we adopt the symplectic midpointmethod The new methods are denoted by LSC schemes Wealso analyze the numerical stability of LSC schemes We givetwo numerical examples to support our theory in Section 4At last we make some conclusions

2 Symplectic Structure of the LSES

Let 119906 = 119901 + 119894119902 The LSES (1) can be written in

119901119905+ 1199021199094 + 119891 (119909) 119902 = 120598119902 ∘ 120594

minus119902119905+ 1199011199094 + 119891 (119909) 119901 = 120598119901 ∘ 120594

(4)

Introducing the variable 119911 = (119901 119902)119879 (4) reads in stochas-

tic symplectic context

119911119905= 119869minus1nabla119911119867(119911) + 120598119869

minus1nabla119911119878 (119911) ∘ 120594 (5)

where

119867(119911) =1

2(1199012

119909119909+ 1199022

119909119909) +

119891 (119909)

2(1199012+ 1199022)

119878 (119911) = minus1

2(1199012+ 1199022)

119869 = (0 1

minus1 0)

(6)

The system satisfies the symplectic conservation law [7 1218]

120596119905= 0 120596 = 119889119901 and 119889119902 (7)

Numerical methods which preserve the discrete symplecticconservation law are called symplectic methods Symplecticmethods have good numerical stability

3 LSC Schemes

31 Compact Scheme Introduce the following uniformmeshgrids

119909119896= 119896ℎ 119896 = 0 1 119873 119905

119899= 119899120591 119899 = 0 1

(8)

where ℎ = 119871119873 and 120591 are spatial and temporal step sizesrespectively Denote the numerical values of 119906(119909

119896 119905119899) at the

nodes (119909119896 119905119899) by 119906

119899

119896 The symbols 119906

119899 and 119906119896mean the

numerical solution vectors at 119905 = 119905119899and 119909 = 119909

119896with compo-

nents 119906119899119896 respectively Furthermore we denote

119906119899+(12)

119896=

119906119899+1

119896+ 119906119899

119896

2 120575

119905119906119899+(12)

119896=

119906119899+1

119896minus 119906119899

119896

120591 (9)

Introducing the following linear operators

A119906119896= 120572119906119896minus1

+ 119906119896+ 120572119906119896+1

B119906119896= 119887

119906119896+3

minus 9119906119896+1

+ 16119906119896minus 9119906119896minus1

+ 119906119896minus3

6ℎ4

+ 119886119906119896+2

minus 4119906119896+1

+ 6119906119896minus 4119906119896minus1

+ 119906119896minus2

ℎ4

(10)

we adopt formula [19]

1205754

119909119906119896= Aminus1B119906119896

(11)

to approximate 1199061199094 which means that

A1205754

119909119906119896= B119906

119896 (12)

By Taylor expansion we can derive a family of fourth-orderschemes with

119886 = 2 (1 minus 120572) 119887 = 4120572 minus 1 (13)

The leading termof the truncation error of themethod is ((7minus26120572)240)(119906

1199098)119896ℎ4 If 119887 = 0 we get a scheme with smaller

stencil A sixth-order scheme is obtained with

120572 =7

26 119886 =

19

13 119887 =

1

13 (14)

Denote two symmetric and cyclic matrices by

119860 =

[[[[[[[[[[[[[[[[

[


120572 1 120572 0 sdot sdot sdot sdot sdot sdot 0

0 120572 1 120572 0 sdot sdot sdot 0

d d d d d

d 120572 1 120572 0

0 0 sdot sdot sdot 0 120572 1 120572

120572 0 sdot sdot sdot sdot sdot sdot 0 120572 1

]]]]]]]]]]]]]]]]

]119873times119873


119861 =1

6ℎ4

[[[[[[[[[[[[[[[[[[

[

1198880

1198881


1198881

1198880

1198881


6119886 1198881

1198880

1198881


119887 6119886 1198881

1198880

1198881


0 119887 6119886 1198881

1198880

1198881

6119886 119887 0 sdot sdot sdot 0

d d d d d d d d d

0 sdot sdot sdot 0 119887 6119886 1198881

1198880

1198881

6119886 119887 0


1198880

1198881

6119886 119887


1198880

1198881

6119886


1198880

1198881

1198881


1198880

]]]]]]]]]]]]]]]]]]

]119873times119873

(15)

where 1198880= 16119887 + 36119886 and 119888


form of (11) is

1205754

119909119906119899= 119860minus1119861119906119899 (16)


(119911119896)119905= 119869minus1nabla119911119867(119911119896) + 120598119869

minus1nabla119911119878 (119911119896) ∘ 120594119896 (17)

where

119867(119911) =119891 (119909119896)

2(1199012

119896+ 1199022

119896) +

1

21205754

119909(1199012

119896+ 1199022

119896)

119878 (119911) = minus1

2(1199012

119896+ 1199022

119896)

(18)


120575119905119911119899+(12)

119896= 119869minus1nabla119911119867(119911119899+(12)

119896) + 120598119869

minus1nabla119911119867(119911119899+(12)

119896)

∘ 120594119899+(12)

119896

(19)


119901119899+1

119896minus 119901119899

119896

120591= minus119891 (119909

119896) 119902119899+(12)

119896minus 1205754

119909119902119899+(12)

119896

+ 120598119902119899+(12)

119896∘ 120594119899+(12)

119896

119902119899+1

119896minus 119902119899

119896

120591= 119891 (119909

119896) 119901119899+(12)

119896+ 1205754

119909119901119899+(12)

119896

minus 120598119901119899+(12)

119896∘ 120594119899+(12)

119896

(20)


1205754

119909119911119899

119896= 120583119911119899

119896 (21)

where

120583 =4(120596 minus 1)

2(3119886 + 2119887 + 119887120596)

3ℎ4 (1 + 2120572120596) 120596 = cos120573ℎ (22)


= 119866119911119899 with

(1 +11988821205912

4)119866 = (

1 minus11988821205912

4minus119888120591

119888120591 1 minus11988821205912

4

) (23)

where 119888 = 120583 + 119891(119909119896) minus 120598 ∘ 120594

119899+(12)





Theorem 2 Let 1199111198992 = ℎsum119896119911119899

119896119911119899




119894A119906119899+1

119896minus 119906119899

119896

120591+B119906

119899+(12)

119896+A119891 (119909

119896) 119906119899+(12)

119896

= 120598A119906119899+(12)

119896∘ 120594119899+(12)

119896

(24)

Multiplying (24) by (119906119899+1119896

minus119906119899


119894120591

2sum

119896

[B119906119899+1

119896119906119899+1

119896minusB119906

119899

119896119906119899

119896]

+119894120591

2sum

119896

A119891 (119909119896) [119906119899+1

119896119906119899+1

119896minus 119906119899

119896119906119899

119896]

minus119894120591120598

2sum

119896

[A119906119899+1

119896∘ 120594119899+(12)

119896119906119899+1

119896minusA119906119899

119896∘ 120594119899+(12)

119896119906119899

119896]

+119894120591

2sum

119896

[B119906119899

119896119906119899+1

119896minusB119906

119899+1

119896119906119899

119896]

+119894120591

2sum

119896

A119891 (119909119896) [119906119899

119896119906119899+1

119896minus 119906119899+1

119896119906119899

119896]

minus119894120591120598

2sum

119896

[A119906119899

119896∘ 120594119899+(12)

119896119906119899+1

119896minusA119906119899+1

119896∘ 120594119899+(12)

119896119906119899

119896]

= sum

119896

[A (119906119899+1

119896minus 119906119899

119896) (119906119899+1

119896minus 119906119899

119896)]

(25)


0

5

10

0

2

4

0

05

1

15

tx

|un k|

(a) 120598 = 005

0

5

10

0

2

4

0

05

1

15

tx

|un k|

(b) 120598 = 01


0 10 20 30 40098

1

102

104

106

108

t

||u||

(a)

0 2 4 6 8 100

05

1

15

2

times10minus12

t

|en Q|

(b)



119864119899= ℎsum

119896

[B119906119899

119896119906119899

119896] + ℎsum

119896

[A119891 (119909119896) 119906119899

119896119906119899

119896]

minusℎ

2120598sum

119896

[A119906119899

119896∘ 120594119899

119896119906119899

119896]

(26)


119864119899+1

minus 119864119899=ℎ

2120598sum

119896

[A119906119899+1

119896∘ 120594119899

119896119906119899+1

119896minusA119906119899

119896∘ 120594119899+1

119896119906119899

119896] (27)

Denote

V119899119896= radic

119887

6

119906119899

119896+3minus 119906119899

119896

ℎ2 V119899

119896= radic

3119887

2

119906119899

119896+1minus 119906119899

119896

ℎ2

119908119899

119896= radic119886

119906119899

119896+2minus 119906119899

119896

ℎ2

119908119899

119896= 2radic119886

119906119899

119896+1minus 119906119899

119896

ℎ2 119910

119899

119896= A119891 (119909

119896) 119906119899

119896

119910119899

119896= A119906

119899

119896∘ 120594119899

119896

⟨119906119899 119910119899⟩ = ℎsum

119896

119906119899

119896119910119899

119896

10038171003817100381710038171199061198991003817100381710038171003817

2

= ⟨119906119899 119906119899⟩

(28)


0 5 10 15 200

001

002

003

004

005

006

007

008

t

|en H|

(a) 119890119899119867

0 5 10 15 200

02

04

06

08

1

t

HnminusH

0

(b) 119867119899 minus 1198670


0

20

40

0

2

4

0

05

1

15

tx

|un k|

(a) 120598 = 005

0

20

40

0

2

4

0

05

1

15

tx

|un k|

(b) 120598 = 01



ℎsum

119896

[B119906119899

119896119906119899

119896] =

1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

(29)

Then

119864119899=1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

+ ⟨119910119899 119906119899⟩ minus

120598

2⟨119910119899 119906119899⟩

(30)





119896= (1radicℎ120591)120594

119899+(12)

119896 where 120594119899+(12)

119896 119896 = 0 1 119873


10038171003817100381710038171199061198991003817100381710038171003817infin = max

1le119896le119873

1003816100381610038161003816119906119899

119896

1003816100381610038161003816 119890119899

119876= 119876119899minus 1198760

119890119899

119867= 119867119899minus 119867119899minus1

(31)


119876and 119890119899


120591 = 001 ℎ = 12058750 and 120598 = 005 01


0 10 20 30 40095

1

105

11

t

||u||

(a)

0 10 20 30 400

02

04

06

08

1

12

times10minus11

t

|en Q|

(b)


0 10 20 30 4001

015

02

025

03

035

04

045

05

t

|en H|

(a) 119890119899119867

0 10 20 30 400

05

1

15

2

t

HnminusH

0

(b) 119867119899 minus 1198670




119906 (119909 0) = exp(1198941205873) cos 2119909

(32)









Consider


119906 (119909 0) = sin2119909(33)


119890119894119905sin2119909






5 Conclusion




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119861 =1

6ℎ4

[[[[[[[[[[[[[[[[[[

[

1198880

1198881


1198881

1198880

1198881


6119886 1198881

1198880

1198881


119887 6119886 1198881

1198880

1198881


0 119887 6119886 1198881

1198880

1198881

6119886 119887 0 sdot sdot sdot 0

d d d d d d d d d

0 sdot sdot sdot 0 119887 6119886 1198881

1198880

1198881

6119886 119887 0


1198880

1198881

6119886 119887


1198880

1198881

6119886


1198880

1198881

1198881


1198880

]]]]]]]]]]]]]]]]]]

]119873times119873

(15)

where 1198880= 16119887 + 36119886 and 119888


form of (11) is

1205754

119909119906119899= 119860minus1119861119906119899 (16)


(119911119896)119905= 119869minus1nabla119911119867(119911119896) + 120598119869

minus1nabla119911119878 (119911119896) ∘ 120594119896 (17)

where

119867(119911) =119891 (119909119896)

2(1199012

119896+ 1199022

119896) +

1

21205754

119909(1199012

119896+ 1199022

119896)

119878 (119911) = minus1

2(1199012

119896+ 1199022

119896)

(18)


120575119905119911119899+(12)

119896= 119869minus1nabla119911119867(119911119899+(12)

119896) + 120598119869

minus1nabla119911119867(119911119899+(12)

119896)

∘ 120594119899+(12)

119896

(19)


119901119899+1

119896minus 119901119899

119896

120591= minus119891 (119909

119896) 119902119899+(12)

119896minus 1205754

119909119902119899+(12)

119896

+ 120598119902119899+(12)

119896∘ 120594119899+(12)

119896

119902119899+1

119896minus 119902119899

119896

120591= 119891 (119909

119896) 119901119899+(12)

119896+ 1205754

119909119901119899+(12)

119896

minus 120598119901119899+(12)

119896∘ 120594119899+(12)

119896

(20)


1205754

119909119911119899

119896= 120583119911119899

119896 (21)

where

120583 =4(120596 minus 1)

2(3119886 + 2119887 + 119887120596)

3ℎ4 (1 + 2120572120596) 120596 = cos120573ℎ (22)


= 119866119911119899 with

(1 +11988821205912

4)119866 = (

1 minus11988821205912

4minus119888120591

119888120591 1 minus11988821205912

4

) (23)

where 119888 = 120583 + 119891(119909119896) minus 120598 ∘ 120594

119899+(12)





Theorem 2 Let 1199111198992 = ℎsum119896119911119899

119896119911119899




119894A119906119899+1

119896minus 119906119899

119896

120591+B119906

119899+(12)

119896+A119891 (119909

119896) 119906119899+(12)

119896

= 120598A119906119899+(12)

119896∘ 120594119899+(12)

119896

(24)

Multiplying (24) by (119906119899+1119896

minus119906119899


119894120591

2sum

119896

[B119906119899+1

119896119906119899+1

119896minusB119906

119899

119896119906119899

119896]

+119894120591

2sum

119896

A119891 (119909119896) [119906119899+1

119896119906119899+1

119896minus 119906119899

119896119906119899

119896]

minus119894120591120598

2sum

119896

[A119906119899+1

119896∘ 120594119899+(12)

119896119906119899+1

119896minusA119906119899

119896∘ 120594119899+(12)

119896119906119899

119896]

+119894120591

2sum

119896

[B119906119899

119896119906119899+1

119896minusB119906

119899+1

119896119906119899

119896]

+119894120591

2sum

119896

A119891 (119909119896) [119906119899

119896119906119899+1

119896minus 119906119899+1

119896119906119899

119896]

minus119894120591120598

2sum

119896

[A119906119899

119896∘ 120594119899+(12)

119896119906119899+1

119896minusA119906119899+1

119896∘ 120594119899+(12)

119896119906119899

119896]

= sum

119896

[A (119906119899+1

119896minus 119906119899

119896) (119906119899+1

119896minus 119906119899

119896)]

(25)


0

5

10

0

2

4

0

05

1

15

tx

|un k|

(a) 120598 = 005

0

5

10

0

2

4

0

05

1

15

tx

|un k|

(b) 120598 = 01


0 10 20 30 40098

1

102

104

106

108

t

||u||

(a)

0 2 4 6 8 100

05

1

15

2

times10minus12

t

|en Q|

(b)



119864119899= ℎsum

119896

[B119906119899

119896119906119899

119896] + ℎsum

119896

[A119891 (119909119896) 119906119899

119896119906119899

119896]

minusℎ

2120598sum

119896

[A119906119899

119896∘ 120594119899

119896119906119899

119896]

(26)


119864119899+1

minus 119864119899=ℎ

2120598sum

119896

[A119906119899+1

119896∘ 120594119899

119896119906119899+1

119896minusA119906119899

119896∘ 120594119899+1

119896119906119899

119896] (27)

Denote

V119899119896= radic

119887

6

119906119899

119896+3minus 119906119899

119896

ℎ2 V119899

119896= radic

3119887

2

119906119899

119896+1minus 119906119899

119896

ℎ2

119908119899

119896= radic119886

119906119899

119896+2minus 119906119899

119896

ℎ2

119908119899

119896= 2radic119886

119906119899

119896+1minus 119906119899

119896

ℎ2 119910

119899

119896= A119891 (119909

119896) 119906119899

119896

119910119899

119896= A119906

119899

119896∘ 120594119899

119896

⟨119906119899 119910119899⟩ = ℎsum

119896

119906119899

119896119910119899

119896

10038171003817100381710038171199061198991003817100381710038171003817

2

= ⟨119906119899 119906119899⟩

(28)


0 5 10 15 200

001

002

003

004

005

006

007

008

t

|en H|

(a) 119890119899119867

0 5 10 15 200

02

04

06

08

1

t

HnminusH

0

(b) 119867119899 minus 1198670


0

20

40

0

2

4

0

05

1

15

tx

|un k|

(a) 120598 = 005

0

20

40

0

2

4

0

05

1

15

tx

|un k|

(b) 120598 = 01



ℎsum

119896

[B119906119899

119896119906119899

119896] =

1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

(29)

Then

119864119899=1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

+ ⟨119910119899 119906119899⟩ minus

120598

2⟨119910119899 119906119899⟩

(30)





119896= (1radicℎ120591)120594

119899+(12)

119896 where 120594119899+(12)

119896 119896 = 0 1 119873


10038171003817100381710038171199061198991003817100381710038171003817infin = max

1le119896le119873

1003816100381610038161003816119906119899

119896

1003816100381610038161003816 119890119899

119876= 119876119899minus 1198760

119890119899

119867= 119867119899minus 119867119899minus1

(31)


119876and 119890119899


120591 = 001 ℎ = 12058750 and 120598 = 005 01


0 10 20 30 40095

1

105

11

t

||u||

(a)

0 10 20 30 400

02

04

06

08

1

12

times10minus11

t

|en Q|

(b)


0 10 20 30 4001

015

02

025

03

035

04

045

05

t

|en H|

(a) 119890119899119867

0 10 20 30 400

05

1

15

2

t

HnminusH

0

(b) 119867119899 minus 1198670




119906 (119909 0) = exp(1198941205873) cos 2119909

(32)









Consider


119906 (119909 0) = sin2119909(33)


119890119894119905sin2119909






5 Conclusion




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

0

2

4

0

05

1

15

tx

|un k|

(a) 120598 = 005

0

5

10

0

2

4

0

05

1

15

tx

|un k|

(b) 120598 = 01


0 10 20 30 40098

1

102

104

106

108

t

||u||

(a)

0 2 4 6 8 100

05

1

15

2

times10minus12

t

|en Q|

(b)



119864119899= ℎsum

119896

[B119906119899

119896119906119899

119896] + ℎsum

119896

[A119891 (119909119896) 119906119899

119896119906119899

119896]

minusℎ

2120598sum

119896

[A119906119899

119896∘ 120594119899

119896119906119899

119896]

(26)


119864119899+1

minus 119864119899=ℎ

2120598sum

119896

[A119906119899+1

119896∘ 120594119899

119896119906119899+1

119896minusA119906119899

119896∘ 120594119899+1

119896119906119899

119896] (27)

Denote

V119899119896= radic

119887

6

119906119899

119896+3minus 119906119899

119896

ℎ2 V119899

119896= radic

3119887

2

119906119899

119896+1minus 119906119899

119896

ℎ2

119908119899

119896= radic119886

119906119899

119896+2minus 119906119899

119896

ℎ2

119908119899

119896= 2radic119886

119906119899

119896+1minus 119906119899

119896

ℎ2 119910

119899

119896= A119891 (119909

119896) 119906119899

119896

119910119899

119896= A119906

119899

119896∘ 120594119899

119896

⟨119906119899 119910119899⟩ = ℎsum

119896

119906119899

119896119910119899

119896

10038171003817100381710038171199061198991003817100381710038171003817

2

= ⟨119906119899 119906119899⟩

(28)


0 5 10 15 200

001

002

003

004

005

006

007

008

t

|en H|

(a) 119890119899119867

0 5 10 15 200

02

04

06

08

1

t

HnminusH

0

(b) 119867119899 minus 1198670


0

20

40

0

2

4

0

05

1

15

tx

|un k|

(a) 120598 = 005

0

20

40

0

2

4

0

05

1

15

tx

|un k|

(b) 120598 = 01



ℎsum

119896

[B119906119899

119896119906119899

119896] =

1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

(29)

Then

119864119899=1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

+ ⟨119910119899 119906119899⟩ minus

120598

2⟨119910119899 119906119899⟩

(30)





119896= (1radicℎ120591)120594

119899+(12)

119896 where 120594119899+(12)

119896 119896 = 0 1 119873


10038171003817100381710038171199061198991003817100381710038171003817infin = max

1le119896le119873

1003816100381610038161003816119906119899

119896

1003816100381610038161003816 119890119899

119876= 119876119899minus 1198760

119890119899

119867= 119867119899minus 119867119899minus1

(31)


119876and 119890119899


120591 = 001 ℎ = 12058750 and 120598 = 005 01


0 10 20 30 40095

1

105

11

t

||u||

(a)

0 10 20 30 400

02

04

06

08

1

12

times10minus11

t

|en Q|

(b)


0 10 20 30 4001

015

02

025

03

035

04

045

05

t

|en H|

(a) 119890119899119867

0 10 20 30 400

05

1

15

2

t

HnminusH

0

(b) 119867119899 minus 1198670




119906 (119909 0) = exp(1198941205873) cos 2119909

(32)









Consider


119906 (119909 0) = sin2119909(33)


119890119894119905sin2119909






5 Conclusion




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 200

001

002

003

004

005

006

007

008

t

|en H|

(a) 119890119899119867

0 5 10 15 200

02

04

06

08

1

t

HnminusH

0

(b) 119867119899 minus 1198670


0

20

40

0

2

4

0

05

1

15

tx

|un k|

(a) 120598 = 005

0

20

40

0

2

4

0

05

1

15

tx

|un k|

(b) 120598 = 01



ℎsum

119896

[B119906119899

119896119906119899

119896] =

1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

(29)

Then

119864119899=1003817100381710038171003817V1198991003817100381710038171003817

2

minus1003817100381710038171003817V1198991003817100381710038171003817

2

+10038171003817100381710038171199081198991003817100381710038171003817

2

minus10038171003817100381710038171199081198991003817100381710038171003817

2

+ ⟨119910119899 119906119899⟩ minus

120598

2⟨119910119899 119906119899⟩

(30)





119896= (1radicℎ120591)120594

119899+(12)

119896 where 120594119899+(12)

119896 119896 = 0 1 119873


10038171003817100381710038171199061198991003817100381710038171003817infin = max

1le119896le119873

1003816100381610038161003816119906119899

119896

1003816100381610038161003816 119890119899

119876= 119876119899minus 1198760

119890119899

119867= 119867119899minus 119867119899minus1

(31)


119876and 119890119899


120591 = 001 ℎ = 12058750 and 120598 = 005 01


0 10 20 30 40095

1

105

11

t

||u||

(a)

0 10 20 30 400

02

04

06

08

1

12

times10minus11

t

|en Q|

(b)


0 10 20 30 4001

015

02

025

03

035

04

045

05

t

|en H|

(a) 119890119899119867

0 10 20 30 400

05

1

15

2

t

HnminusH

0

(b) 119867119899 minus 1198670




119906 (119909 0) = exp(1198941205873) cos 2119909

(32)









Consider


119906 (119909 0) = sin2119909(33)


119890119894119905sin2119909






5 Conclusion




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40095

1

105

11

t

||u||

(a)

0 10 20 30 400

02

04

06

08

1

12

times10minus11

t

|en Q|

(b)


0 10 20 30 4001

015

02

025

03

035

04

045

05

t

|en H|

(a) 119890119899119867

0 10 20 30 400

05

1

15

2

t

HnminusH

0

(b) 119867119899 minus 1198670




119906 (119909 0) = exp(1198941205873) cos 2119909

(32)









Consider


119906 (119909 0) = sin2119909(33)


119890119894119905sin2119909






5 Conclusion




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














5 Conclusion




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Symplectic Schemes for Linear Stochastic ...downloads.hindawi.com/journals/aaa/2014/427023.pdf · Symplectic Schemes for Linear Stochastic Schrödinger Equations