Open AccessRigorous numerics for ODEs using Chebyshev series and domain decomposition
Author(s) -
Jan Bouwe van den Berg,
Ray Sheombarsing
Publication year - 2021
Publication title -
journal of computational dynamics
Language(s) - English
Resource type - Journals
eISSN - 2158-2505
pISSN - 2158-2491
DOI - 10.3934/jcd.2021015
Subject(s) - chebyshev filter , ode , mathematics , chebyshev pseudospectral method , chebyshev equation , chebyshev iteration , chebyshev polynomials , chebyshev nodes , approximation theory , series (stratigraphy) , fixed point , domain decomposition methods , domain (mathematical analysis) , mathematical analysis , finite element method , orthogonal polynomials , classical orthogonal polynomials , paleontology , physics , biology , thermodynamics
In this paper we present a rigorous numerical method for validating analytic solutions of nonlinear ODEs by using Chebyshev-series and domain decomposition. The idea is to define a Newton-like operator, whose fixed points correspond to solutions of the ODE, on the space of geometrically decaying Chebyshev coefficients, and to use the so-called radii-polynomial approach to prove that the operator has an isolated fixed point in a small neighborhood of a numerical approximation. The novelty of the proposed method is the use of Chebyshev series in combination with domain decomposition. In particular, a heuristic procedure based on the theory of Chebyshev approximations for analytic functions is presented to construct efficient grids for validating solutions of boundary value problems. The effectiveness of the proposed method is demonstrated by validating long periodic and connecting orbits in the Lorenz system for which validation without domain decomposition is not feasible.