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Validated continuation over large parameter ranges for equilibria of PDEs

Author

Listed:
  • Gameiro, Marcio
  • Lessard, Jean-Philippe
  • Mischaikow, Konstantin

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.

Suggested Citation

  • Gameiro, Marcio & Lessard, Jean-Philippe & Mischaikow, Konstantin, 2008. "Validated continuation over large parameter ranges for equilibria of PDEs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(4), pages 1368-1382.
    • Handle: RePEc:eee:matcom:v:79:y:2008:i:4:p:1368-1382
    DOI: 10.1016/j.matcom.2008.03.014
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