On the Time Splitting Spectral Method for the Complex Ginzburg-Landau Equation in the Large Time and Space Scale Limit

We are interested in the numerical approximation of the complex Ginzburg-Landau equation in the large time and space limit. There are two interesting regimes in this problem, one being the large space time limit, and one being the nonlinear Schrodinger limit. These limits have been studied analytically in, for example, T. Colin and A. Soyeur, Asymptot. Anal., 13 (1996), pp. 361-372; F. H. Lin, Comm. Pure Appl. Math., 51 (1998), pp. 385-441; F. H. Lin and J. X. Xin, Comm. Math. Phys., 200 (1999), pp. 249-274. We study a time splitting spectral method for this problem. In particular, we are interested in whether such a scheme is asymptotic preserving (AP) with respect to these two limits. Our results show that the scheme is AP for the first limit but not the second one. For the large space time limit, our numerical experiments show that the scheme can capture the correct physical behavior without resolving the small scale dynamics, even for transitional problems, where small and large scales coexist.

[1]  John Whitehead,et al.  Finite bandwidth, finite amplitude convection , 1969, Journal of Fluid Mechanics.

[2]  Lee A. Segel,et al.  Distant side-walls cause slow amplitude modulation of cellular convection , 1969, Journal of Fluid Mechanics.

[3]  Lee A. Segel,et al.  Non-linear wave-number interaction in near-critical two-dimensional flows , 1971, Journal of Fluid Mechanics.

[4]  John Whitehead,et al.  Review of the Finite Bandwidth Concept , 1971 .

[5]  D. Pathria,et al.  Pseudo-spectral solution of nonlinear Schro¨dinger equations , 1990 .

[6]  John C. Neu,et al.  Vortices in complex scalar fields , 1990 .

[7]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[8]  Andrew M. Stuart,et al.  Discrete Gevrey regularity attractors and uppers–semicontinuity for a finite difference approximation to the Ginzburg–Landau equation , 1995 .

[9]  Thierry Colin,et al.  Some singular limits for evolutionary Ginzburg–Landau equations , 1996 .

[10]  Alessandro Torcini,et al.  Studies of phase turbulence in the one-dimensional complex Ginzburg-Landau equation , 1996, chao-dyn/9608003.

[11]  G. Lord.  Attractors and Inertial Manifolds for Finite-Difference Approximations of the Complex Ginzburg--Landau Equation , 1997 .

[12]  F. Lin.  Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds , 1998 .

[13]  Stefan C. Müller,et al.  Evolution of spontaneous structures in dissipative continuous systems , 1998 .

[14]  Shi Jin,et al.  Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations , 1999, SIAM J. Sci. Comput..

[15]  J. Xin,et al.  On the Incompressible Fluid Limit and the Vortex Motion Law of the Nonlinear Schrödinger Equation , 1999 .

[16]  François Golse,et al.  The Convergence of Numerical Transfer Schemes in Diffusive Regimes I: Discrete-Ordinate Method , 1999 .

[17]  Ansgar Jüngel,et al.  A Nonstiff Euler Discretization of the Complex Ginzburg-Landau Equation in One Space Dimension , 2000, SIAM J. Numer. Anal..

[18]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

[19]  Christophe Besse,et al.  Order Estimates in Time of Splitting Methods for the Nonlinear Schrödinger Equation , 2002, SIAM J. Numer. Anal..

[20]  Arjen Doelman.  UvA-DARE ( Digital Academic Repository ) Finite-dimensional models of the Ginzburg-Landau equation Doelman , 2002 .

[21]  P. Markowich,et al.  On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime , 2002 .

[22]  Shi Jin,et al.  Numerical Study of Time-Splitting Spectral Discretizations of Nonlinear Schrödinger Equations in the Semiclassical Regimes , 2003, SIAM J. Sci. Comput..

[23]  P. Markowich,et al.  Numerical simulation of a generalized Zakharov system , 2004 .

[24]  Chunxiong Zheng,et al.  A time-splitting spectral scheme for the Maxwell-Dirac system , 2005, 1205.0368.

[25]  Qiang Du,et al.  Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation , 2007, European Journal of Applied Mathematics.

[26]  Qiang Du,et al.  The Dynamics and Interaction of Quantized Vortices in the Ginzburg-Landau-Schrödinger Equation , 2007, SIAM J. Appl. Math..

[27]  Pierre Degond,et al.  An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit , 2007, J. Comput. Phys..

[28]  Christian Lubich,et al.  On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations , 2008, Math. Comput..