Abstract

A series of new methods based on the Runge-Kutta (RK) formula are proposed, which not only retain the merit of RK methods, in that the adaptive stepsize is easily implemented, but also dramatically decrease the error under the same conditions. Based on the new methods, an effective shooting algorithm is also proposed. A two-point boundary value problem for the Raman amplifier propagation equations is solved using the proposed algorithm. Our algorithm markedly increases the simulating speed for Raman amplifier propagation equations, as well as improves the accuracy, compared to the traditional algorithm.

© 2003 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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IEEE J. Quantum Electron

X. M. Liu and B. Lee, ???A series of fast and accurate algorithms and their applications in fiber transmission systems,??? (submitted to IEEE J. Quantum Electron.)

IEEE Photon. Technol. Lett

N. Kikuchi, K. K. Wong, K. Uesaka, K. Shimizu, S. Yam, E. S. Hu, M. Marhic, and L. G. Kazovsky, ???Novel in-service wavelength-band upgrade scheme for fiber Raman amplifier,??? IEEE Photon. Technol. Lett. 15, 27???29 (2003).
[CrossRef]

IEEE Photon. Technol. Lett.

X. M. Liu, H. Y Zhang, and Y. L Guo, ???A novel method for Raman amplifier propagation equations,??? IEEE Photon. Technol. Lett. 15, 392-394 (2003).
[CrossRef]

A. Carena, V. Curri, and P. Poggiolini, ???On the optimization of hybrid Raman/erbium-doped fiber amplifiers,??? IEEE Photon. Technol. Lett. 13, 1170-1172 (2001).
[CrossRef]

J. Lightwave Technol.

Math. Comp.

R. Weiss, ???The application of implicit Runge-Kutta and collocation methods to boundary value problems,??? Math. Comp. 28, 449???464 (1974).

Opt. Commun.

Q. H. Mao, J. S. Wang, X. H. Sun, and M. D. Zhang, ???A theoretical analysis of amplification characteristics of bi-directional erbium-doped fiber amplifiers with single erbium-doped fiber,??? Opt. Commun. 159, 149-157 (1999).
[CrossRef]

SIAM J. Sci. Comput.

W. H. Enright and P. H. Muir, ???Runge-Kutta software with defect control for boundary value ODEs,??? SIAM J. Sci. Comput. 17, 479???497 (1996).
[CrossRef]

Other

X. M. Liu and B. Lee, ???A fast and stable method for Raman amplifier propagation equations,??? (to be submitted)

U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations (SIAM, Philadelphia, 1998).
[CrossRef]

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer-Verlag, New York, 2000).

B. F. Plybon, An Introduction to Applied Numerical Analysis (PWS-KENT Publishing Company, Boston, 1992), pp. 428-441

J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering (Second edition, Prentice Hall, New Jersey, 1992), pp. 464-475.

U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (SIAM, Philadelphia, 1995).
[CrossRef]

P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two Point Boundary Value Problems (Academic Press, New York, 1968).

L. F. Shampine, Numerical Solution of Ordinary Differential Equations (Chapman & Hall, New York, 1994).

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Figures (4)

Fig. 1.
Fig. 1.

Relationship of relative error with t, for (a) the fourth-order RK method, and (b) the fourth-order ORK method.

Fig. 2.
Fig. 2.

Calculation procedure for the shooting algorithm based on the fourth-order ORK method, where (a) for y 1(t), (b) for y 2(t), and (c) for the relative error of y 1(t) and y 2(t) in the last iteration.

Fig. 3.
Fig. 3.

Raman gain spectrum gR v) of the fiber, which was used in the simulation.

Fig. 4.
Fig. 4.

Iteration procedure and numerical results for a distributed pump Raman amplifier, for (a) iteration procedures of a pump, (b) iteration procedures of a signal, and (c) results of all pumps and signals at the last iteration. The calculating procedure is from z=L to z=0.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

d y d t = f ( t , y ) ,
y ( t = t 0 ) = y 0 .
y n + 1 = y n + i = 1 m w i k i ,
k i = h f ( t n + c i , y n + j = 1 m d ij k j ) , i = 1 , 2 , m .
d z d t = g ( t , exp ( z ) ) .
z n + 1 = z n + i = 1 m w i q i ,
q i = h g ( t n + c i , exp ( z n ) ( 1 + j = 1 m d ij q j ) ) , i = 1 , 2 , m .
y n + 1 = y n exp ( i = 1 m w i q i ) ,
q i = h g ( t n + c i , y n + y n j = 1 m d ij q j ) , i = 1 , 2 , m .
y n + 1 = y n exp [ ( q 1 + 2 q 2 + 2 q 3 + q 4 ) h 6 ] ,
q 1 = g ( t n , y n ) ,
q 2 = g ( t n + h 2 , y n + y n q 1 h 2 ) ,
q 3 = g ( t n + h 2 , y n + y n q 2 h 2 ) ,
q 4 = g ( t n + h , y n + y n q 3 h ) ,
g ( t n ) = f ( t n ) y n .
ε = C h 5 [ In ( y ) ] ( 5 ) ( ξ ) .
ε = C h 5 y ( 5 ) ( ξ ) .
dy dt = y ( t 3 200 + t 2 50 + t ) , y ( 0 ) = 1 , and t [ 0 , 3.6 ] .
y = exp ( t 4 800 + t 3 150 + t 2 2 ) .
d y 1 ( t ) d t = f 1 ( t , y 1 , y 2 ) ,
d y 2 ( t ) d t = f 2 ( t , y 1 , y 2 ) ,
y 1 ( a ) = α ,
y 2 ( b ) = β .
A j + 1 = { A j 1 + ( β β j 1 ) ( A j A j 1 ) ( β j β j 1 ) , ( if j > 1 ) A j β β j , ( if j = 1 )
± d P i d z = [ α i + j = 1 i 1 g R ( v j v i ) Γ A eff P j j = i + 1 m v i v j g R ( v i v j ) Γ A eff P j ] P i , ( i = 1 , 2 , , m ) ,

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