Construction of approximate inertial manifold by decimation of collocation equations of distributed parameter systems

Abstract A collocation method is adopted as a numerical framework to develop approximate inertial manifolds (AIMs) in the case of partial differential problems (e.g. reaction/diffusion models) containing non-polynomial nonlinearities. The spatial discretization, based on the collocation approach, is the starting point for the alternative construction of AIMs by means of a renormalization/decimation approach naturally derived from the incremental unknown method developed by Temam in a finite difference framework.

[1]  G. Sell,et al.  On the computation of inertial manifolds , 1988 .

[2]  J. Villadsen,et al.  Solution of differential equation models by polynomial approximation , 1978 .

[3]  Wolf,et al.  Controlling spatiotemporal patterns on a catalytic wafer. , 1994, Physical review letters.

[4]  J. Yeomans,et al.  Statistical mechanics of phase transitions , 1992 .

[5]  Arnéodo,et al.  Crisis-induced intermittent bursting in reaction-diffusion chemical systems. , 1992, Physical review letters.

[6]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[7]  Samuel Paolucci,et al.  A multilevel wavelet collocation method for solving partial differential equations in a finite domain , 1995 .

[8]  M. Giona,et al.  Wavelet-like collocation method for finite-dimensional reduction of distributed systems , 2000 .

[9]  A. N. Sharkovskiĭ.  Dynamic systems and turbulence , 1989 .

[10]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[11]  Lawrence Sirovich,et al.  Trends and perspectives in applied mathematics , 1994 .

[12]  Roger Temam,et al.  Inertial manifolds and multigrid methods , 1990 .

[13]  Ioannis G. Kevrekidis,et al.  Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis , 1996 .

[14]  J. C. Robinson,et al.  Finite-dimensional behavior in dissipative partial differential equations. , 1995, Chaos.

[15]  L. Sirovich.  Chaotic dynamics of coherent structures , 1989 .

[16]  Andreas Rieder,et al.  Wavelets: Theory and Applications , 1997 .

[17]  Roger Temam.  Applications of inertial manifolds to scientific computing: a new insight in multilevel methods , 1994 .

[18]  Hsueh-Chia Chang,et al.  Low-dimensional spatio-temporal thermal dynamics on nonuniform catalytic surfaces , 1993 .

[19]  L. Sirovich,et al.  Optimal low-dimensional dynamical approximations , 1990 .

[20]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[21]  J. Hyman,et al.  The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems , 1986 .

[22]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .