Weighted least squares collocation methods

• DOI: 10.1016/j.apnum.2024.05.017
• Publication Date: 2024-05-24T16:18:07Z
• Authors: Luigi Brugnano; Felice Iavernaro; Ewa B. Weinmüller
• Keywords: Hamiltonian boundary value method; Least squares collocation method; Least squares collocation methods; Gauss-Legendre collocation methods; Line integral methods; Hamiltonian boundary value methods; Gauss-Legendre collocation method; Line integral methods; 510

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 0(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.