Weighted least squares collocation methods

We consider overdetermined collocation methods and propose a weighted least squares approach to derive a numerical solution. The discrete problem requires the evaluation of the Jacobian of the vector field which, however, appears in a O(h) term, h being the stepsize. We show that, by neglecting this...

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Bibliographic Details
Published in:Applied numerical mathematics 2024-09, Vol.203, p.113-128
Main Authors: Brugnano, Luigi, Iavernaro, Felice, Weinmüller, Ewa B.
Format: Article
Language:English
Subjects:
Gauss-Legendre collocation methods
Hamiltonian boundary value methods
Least squares collocation methods
Line integral methods
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Summary:We consider overdetermined collocation methods and propose a weighted least squares approach to derive a numerical solution. The discrete problem requires the evaluation of the Jacobian of the vector field which, however, appears in a O(h) term, h being the stepsize. We show that, by neglecting this infinitesimal term, the resulting scheme becomes a low-rank Runge–Kutta method. Among the possible choices of the weights distribution, we analyze the one based on the quadrature formula underlying the collocation conditions. A few numerical illustrations are included to better elucidate the potential of the method.
ISSN:0168-9274
DOI:10.1016/j.apnum.2024.05.017