Validated continuation over large parameter ranges for equilibria of PDEs

Authors:
Marcio Gameiro;Jean-Philippe Lessard;Konstantin Mischaikow
Affiliations:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, USA;Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, USA and Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, The Netherlands;Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, USA
Venue:
Mathematics and Computers in Simulation
Year:
2008

Citing 3
Cited 1

The fast Fourier transform and its applications

The fast Fourier transform and its applications
An efficient estimate based on FFT in topological verification method

Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
Validated Continuation for Equilibria of PDEs

SIAM Journal on Numerical Analysis

Efficient and Generic Algorithm for Rigorous Integration Forward in Time of dPDEs: Part I

Journal of Scientific Computing

Quantified Score

Hi-index	0.00

Visualization

Abstract

Validated continuation was introduced [S. Day, J.-P. Lessard, K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, in press] as means of checking that the classical continuation method applied to a Galerkin projection of a PDE provides a locally unique equilibrium to the PDE of interest. In this paper, we extend the numerical technique to include a parameter that leads to better bounds on the errors associated with the Galerkin truncation. We test this method on the Swift-Hohenberg and Allen-Cahn equations on one-dimensional domains. For the first equation, we find no numerical obstructions to the validated continuation technique. This is not the case for the Allen-Cahn equation.