LOW-DIMENSIONAL MODELS OF THE GINZBURG–LANDAU EQUATION ∗ FRANK KWASNIOK † SIAM J. APPL. MATH. c  2001 Society for Industrial and Applied Mathematics Vol. 61, No. 6, pp. 2063–2079 Abstract. Low-dimensional numerical approximations for two boundary value problems of the complex Ginzburg–Landau equation in a chaotic regime are constructed. Spatial structures called principal interaction patterns are extracted from the system according to a nonlinear variational principle and used as basis functions in a Galerkin approximation. The dynamical description in terms of principal interaction patterns requires fewer modes than more conventional approaches using Fourier modes or Karhunen–Lo` eve modes as basis functions. Key words. low-dimensional numerical approximations, optimal basis functions, nonlinear variational principle, Ginzburg–Landau equation AMS subject classifications. 35A40, 65M60, 65K10, 35Q35 PII. S0036139900368212 1. Introduction. Given the well-known fact that solutions to dissipative partial differential equations are often attracted to manifolds of relatively low dimension, the construction of minimal dynamical approximations capturing the principal properties of the long-term behavior of the complete system is an interesting task. A class of frequently used numerical approximation schemes is given by the Galerkin methods. The efficiency of such Galerkin schemes, i.e., the number of degrees of freedom re- quired to capture the long-term dynamics of the partial differential equation, depends crucially on a proper choice of the basis functions. Traditionally, the basis functions are eigenfunctions of a suitably chosen linear differential operator, commonly Fourier modes. However, such Fourier–Galerkin methods cannot be expected to be very effi- cient since Fourier modes are completely general. A dynamical description based on modes adapted to the particular system under consideration should be more adequate when searching for a minimal model. The Karhunen–Lo` eve (KL) decomposition (also referred to as principal compo- nent analysis, empirical orthogonal function analysis, or proper orthogonal decompo- sition) has occasionally been used as a tool to arrive at low-dimensional dynamical approximations of partial differential equations [16, 3, 15, 10, 11, 12, 13]. The KL modes allow for an optimal representation of an attractor in phase space in a mean least-squares sense with a given number of modes. They are easily obtained as the eigenfunctions of the covariance tensor of the system. The KL approach has been extended to the Sobolev eigenfunctions [7]. However, the optimality criterion defining the KL modes and Sobolev eigenfunctions, respectively, does not refer to the time evolution of the truncated system obtained when projecting the partial differential equation onto these modes. As has been pointed out by Hasselmann [6], a methodol- ogy referring simultaneously to both spatial and temporal features of the system by taking into account the dynamics of the reduced model in order to define the basis functions may be even more efficient. The proposition of Hasselmann has been illus- trated and applied to a geophysical fluid system by Kwasniok [8]. It has also been ∗ Received by the editors February 16, 2000; accepted for publication (in revised form) November 15, 2000; published electronically April 24, 2001. http://www.siam.org/journals/siap/61-6/36821.html † Leibniz-Institut f¨ ur Atmosph¨arenphysik, Schlossstraße 6, 18225 K¨ uhlungsborn, Germany (kwasniok@iap-kborn.de). 2063