A novel method for simulating the complex Ginzburg-Landau equation
Authors:
Daniel Goldman and Lawrence Sirovich
Journal:
Quart. Appl. Math. 53 (1995), 315-333
MSC:
Primary 35Q55; Secondary 65M12, 76E30, 76F99
DOI:
https://doi.org/10.1090/qam/1330655
MathSciNet review:
MR1330655
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We present a split-step method for integration of the complex Ginzburg-Landau equation in any number of spatial dimensions. The novel aspect of the method lies in the fact that each portion of the splitting is explicitly integrable. This leads to an extremely fast, stable, and efficient procedure. A comparison is made with spectral and pseudospectral procedures which have appeared in the literature.
M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon, and M. Gisselfalt, Hard turbulence in a finite dimensional dynamical system?, Phys. Lett. A 142, 349–356 (1989)
M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon, and M. Gisselfalt, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D 44, 421–444 (1990)
M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon, and M. Gisselfalt, Errata, Phys. Lett. A 145, 476 (1990)
A. J. Bernoff, Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation, Phys. D 30, 363–381 (1988)
P. Coullet, L. Gil, and J. Lega, A form of turbulence associated with defects, Phys. D 37, 91–103 (1989)
E. Forest and R. D. Ruth, Fourth-order symplectic integration, Phys. D 43, 105–117 (1990)
A. V. Gaponov-Grekhov and M. I. Rabinovich, Theory of dynamic turbulence (A. H. Luther, ed.), Advances in Theoretical Physics, Pergamon Press, New York, 1988, pp. 177–192
R. H. Hardin and F. D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Rev. 15, 423 (1973)
A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys. Lett. 23, 142–144 (1973)
B. M. Herbst and F. Varadi, Symplectic methods for the nonlinear Schrödinger equation, preprint, 1992
W. H. Louisell, M. Lax, G. P. Agrawal, and H. W. Gatzke, Simultaneous forward and backward integration for standing waves in a resonator, Appl. Optics 18, 2730–2731 (1979)
R. I. McLachlan and P. Atela, The accuracy of symplectic integrators, Nonlinearity 5, 541–562 (1992)
J. V. Moloney, M. R. Belic, and H. M. Gibbs, Calculation of transverse effects in optical bistability using fast Fourier transform techniques, Optics Comm. 41, 379–382 (1982)
S.-C. Sheng and A. E. Siegman, Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction, Phys. Rev. A 21, 599–606 (1980)
L. Sirovich, J. D. Rodriguez, and B. Knight, Two boundary value problems for the Ginzburg-Landau equation, Phys. D 43, 63–76 (1990)
T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys. 55, 203–230 (1984)
F. Tappert, Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method, Nonlinear Wave Motion (A. C. Newell, ed.), Amer. Math. Soc., Providence, RI, 1974, pp. 215–216
F. D. Tappert and C. N. Judice, Recurrence of nonlinear ion acoustic waves, Phys. Rev. Lett. 29, 1308–1311 (1972)
J. A. C. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 23, 485–507 (1986)
H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A 150, 262–268 (1990)
M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon, and M. Gisselfalt, Hard turbulence in a finite dimensional dynamical system?, Phys. Lett. A 142, 349–356 (1989)
M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon, and M. Gisselfalt, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D 44, 421–444 (1990)
M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon, and M. Gisselfalt, Errata, Phys. Lett. A 145, 476 (1990)
A. J. Bernoff, Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation, Phys. D 30, 363–381 (1988)
P. Coullet, L. Gil, and J. Lega, A form of turbulence associated with defects, Phys. D 37, 91–103 (1989)
E. Forest and R. D. Ruth, Fourth-order symplectic integration, Phys. D 43, 105–117 (1990)
A. V. Gaponov-Grekhov and M. I. Rabinovich, Theory of dynamic turbulence (A. H. Luther, ed.), Advances in Theoretical Physics, Pergamon Press, New York, 1988, pp. 177–192
R. H. Hardin and F. D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Rev. 15, 423 (1973)
A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys. Lett. 23, 142–144 (1973)
B. M. Herbst and F. Varadi, Symplectic methods for the nonlinear Schrödinger equation, preprint, 1992
W. H. Louisell, M. Lax, G. P. Agrawal, and H. W. Gatzke, Simultaneous forward and backward integration for standing waves in a resonator, Appl. Optics 18, 2730–2731 (1979)
R. I. McLachlan and P. Atela, The accuracy of symplectic integrators, Nonlinearity 5, 541–562 (1992)
J. V. Moloney, M. R. Belic, and H. M. Gibbs, Calculation of transverse effects in optical bistability using fast Fourier transform techniques, Optics Comm. 41, 379–382 (1982)
S.-C. Sheng and A. E. Siegman, Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction, Phys. Rev. A 21, 599–606 (1980)
L. Sirovich, J. D. Rodriguez, and B. Knight, Two boundary value problems for the Ginzburg-Landau equation, Phys. D 43, 63–76 (1990)
T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys. 55, 203–230 (1984)
F. Tappert, Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method, Nonlinear Wave Motion (A. C. Newell, ed.), Amer. Math. Soc., Providence, RI, 1974, pp. 215–216
F. D. Tappert and C. N. Judice, Recurrence of nonlinear ion acoustic waves, Phys. Rev. Lett. 29, 1308–1311 (1972)
J. A. C. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 23, 485–507 (1986)
H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A 150, 262–268 (1990)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35Q55,
65M12,
76E30,
76F99
Retrieve articles in all journals
with MSC:
35Q55,
65M12,
76E30,
76F99
Additional Information
Article copyright:
© Copyright 1995
American Mathematical Society
