| Abstract: |
| In this talk, we present an improved integrator for the study of solutions to initial and boundary value problems for semilinear parabolic PDEs. Our method is based on a non-autonomous semigroup approach, in which the solution map of the linearized PDE along a numerically computed approximate solution is rigorously controlled. In particular, the use of a multi-stepping scheme, based on a new construction of the multi-step propagator via direct backward iteration, allows us to integrate over much longer time intervals. For each time step, we split the dynamics into a signed finite-dimensional bare propagator and a finite-tail coupling correction. The bare propagators are composed at the signed matrix level, which preserves cancellation effects and significantly reduces the wrapping effect. We also present examples for both initial value problems and boundary value problems, including long-time rigorous integration of the Ohta--Kawasaki equation. |
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