American Institute of Mathematical Sciences

Advanced
November  2002, 8(4): 815-833. doi: 10.3934/dcds.2002.8.815

Computing inertial manifolds

James C. Robinson 1,

1. 

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received  March 2001 Revised  March 2002 Published  July 2002

This paper discusses two numerical schemes that can be used to approximate inertial manifolds whose existence is given by one of the standard methods of proof. The methods considered are fully numerical, in that they take into account the need to interpolate the approximations of the manifold between a set of discrete gridpoints. As all the discretisations are refined the approximations are shown to converge to the true manifold.
Keywords: approximate inertial manifolds., Inertial manifolds.
Mathematics Subject Classification: 35B40, 35K22, 65M1.
Citation: James C. Robinson. Computing inertial manifolds. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 815-833. doi: 10.3934/dcds.2002.8.815
[1]

Ricardo Rosa. Approximate inertial manifolds of exponential order. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 421-448. doi: 10.3934/dcds.1995.1.421
[2]

José M. Arrieta, Esperanza Santamaría. Estimates on the distance of inertial manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 3921-3944. doi: 10.3934/dcds.2014.34.3921
[3]

James C. Robinson. Inertial manifolds with and without delay. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 813-824. doi: 10.3934/dcds.1999.5.813
[4]

Rolf Bronstering. Some computational aspects of approximate inertial manifolds and finite differences. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 417-454. doi: 10.3934/dcds.1996.2.417
[5]

A. Debussche, R. Temam. Some new generalizations of inertial manifolds. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 543-558. doi: 10.3934/dcds.1996.2.543
[6]

Boling Guo, Bixiang Wang. Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 455-466. doi: 10.3934/dcds.1996.2.455
[7]

Oscar P. Manley. Some physical considerations attendant to the approximate inertial manifolds for Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 585-593. doi: 10.3934/dcds.1996.2.585
[8]

Olivier Goubet. Approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 503-530. doi: 10.3934/dcds.1997.3.503
[9]

A. V. Rezounenko. Inertial manifolds with delay for retarded semilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 829-840. doi: 10.3934/dcds.2000.6.829
[10]

Peter Brune, Björn Schmalfuss. Inertial manifolds for stochastic pde with dynamical boundary conditions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 831-846. doi: 10.3934/cpaa.2011.10.831
[11]

Changbing Hu, Kaitai Li. A simple construction of inertial manifolds under time discretization. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 531-540. doi: 10.3934/dcds.1997.3.531
[12]

Vladimir V. Chepyzhov, Anna Kostianko, Sergey Zelik. Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1115-1142. doi: 10.3934/dcdsb.2019009
[13]

Norbert Koksch, Stefan Siegmund. Feedback control via inertial manifolds for nonautonomous evolution equations. Communications on Pure & Applied Analysis, 2011, 10 (3) : 917-936. doi: 10.3934/cpaa.2011.10.917
[14]

L. Dieci, M. S Jolly, Ricardo Rosa, E. S. Van Vleck. Error in approximation of Lyapunov exponents on inertial manifolds: The Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 555-580. doi: 10.3934/dcdsb.2008.9.555
[15]

Cung The Anh, Le Van Hieu, Nguyen Thieu Huy. Inertial manifolds for a class of non-autonomous semilinear parabolic equations with finite delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 483-503. doi: 10.3934/dcds.2013.33.483
[16]

Anna Kostianko, Sergey Zelik. Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2069-2094. doi: 10.3934/cpaa.2015.14.2069
[17]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116
[18]

Jesenko Vukadinovic. Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 327-341. doi: 10.3934/dcds.2011.29.327
[19]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017
[20]

Pierre Degond, Hailiang Liu. Kinetic models for polymers with inertial effects. Networks & Heterogeneous Media, 2009, 4 (4) : 625-647. doi: 10.3934/nhm.2009.4.625

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences