A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations (Q2927841)

The paper is devoted to the discussion of an \(hp\) collocation method for the numerical solution to nonlinear Volterra integral equations of the form \(u(t) = g(t) + \int_0^t K(t,s) G(u(s)) ds\) where, in particular, the nonlinearity is assumed to satisfy a Lipschitz condition and to be independent of the variables \(t\) and \(s\). The method uses an arbitrary, not necessarily uniform, partition of the basic interval and a polynomial collocation using Gauss-Legendre points on each subinterval. It is possible to choose interpolation polynomials of different degree on different subintervals. Error estimates are provided under the assumption that the kernel and the solution have a certain number of continuous derivatives.