Validated continuation over large parameter ranges for equilibria of PDEs

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.