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Efficient and generic algorithm for rigorous integration forward in time of dPDEs. I. (English) Zbl 1296.65138

Summary: We propose an efficient and generic algorithm for rigorous integration forward in time of partial differential equations written in the Fourier basis. By rigorous integration we mean a procedure which operates on sets and return sets which are guaranteed to contain the exact solution. The presented algorithm generates, in an efficient way, normalized derivatives which are used by the Lohner algorithm to produce a rigorous bound. The algorithm has been successfully tested on several partial differential equations (PDEs) including the Burgers equation, the Kuramoto-Sivashinsky equation, and the Swift-Hohenberg equation. The problem of rigorous integration in time of partial differential equations is a problem of large computational complexity and efficient algorithms are required to deal with PDEs on higher dimensional domains, like the Navier-Stokes equation. Technicalities regarding the various optimization techniques implemented in the software used in this paper will be reported elsewhere.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
65G20 Algorithms with automatic result verification
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms
65T50 Numerical methods for discrete and fast Fourier transforms

Software:

FADBAD++; CAPD; Taylor
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References:

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