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Session II.1 - Computational Dynamics

Thursday, June 15, 14:00 ~ 14:30

Validated integration of semilinear parabolic PDEs

Maxime Breden

Ecole polytechnique, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this talk, we present a computer-assisted proof methodology to perform rigorous time integration for semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variations of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition in time, we then finish the proof with a Newton-Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift-Hohenberg equation, the Ohta-Kawasaki equation and the Kuramoto–Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.

Joint work with Jan Bouwe van den Berg (VU Amsterdam) and Ray Sheombarsing (VU Amsterdam).

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