Validated Continuation for Equilibria of PDEs

Abstract

One of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor-corrector continuation techniques. In the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation. Using the Cahn-Hilliard and Swift-Hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone.

Computing (Un)stable Manifolds with Validated Error Bounds: Non-resonant and Resonant Spectra

“…A proof of this lemma can be found, e.g., in Yamamoto (1998), Day et al (2007) and Hungria et al (2016).…”

“…We stress that in this way the radius of the ball on which we obtain contractivity is a variable for which we solve. This is an essential difference of the method in Day et al (2007) compared to classical Newton-Kantorovich-type arguments. Let us assemble the ingredients to define the zero finding problem.…”

“…Our goal is to compute a numerical approximation of P together with rigorous bounds on the approximation error and its range of validity B ν by using the method presented in Day et al (2007). This amounts to first formulating an equivalent zero finding problem on an appropriate Banach space.…”

“…The zero of f corresponds to a parametrization of the stable manifold, a fact we still need to make more precise in due course. We follow the setup of Day et al (2007) and Hungria et al (2016) and derive an equivalent fixed point operator, whose contractivity on a ball around an approximate solution we establish using the so-called radii polynomials. Similar approaches have been used in previous works such as Eckmann et al (1984), Arioli and Koch (2010) and Arioli and Koch (2015) and earlier in Lanford (1982).…”

“…Once these givens are fixed the roots of the radii polynomials yield not only existence and uniqueness results, but also tight error estimates and isolation bounds for the problem. Radii polynomial methods have been applied successfully to a number of problems in dynamical systems, partial differential equations and delay equations, and we refer the interested reader to Day et al (2007), Breden et al (2013), van den Berg et al (2010), van den Berg et al (2011), Hungria et al (2016) and van den Berg et al (2015) for more discussion and references.…”