The stability function for multistep collocation methods

Abstract

Summary. C-polynomials for rational approximation to the exponential function was introduced by N~rsett [7] to study stability properties of one-step methods. For one-step collocation methods the C-polynomial has a very simple form. In this paper we study C-polynomials for multistep collocation methods and obtain results that generalize those in the one-step case, and provide a way to analyze linear stability of such methods.

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

“…Recently, we have proposed multistep collocation methods [13][14][15][16] and two step almost collocation methods [13,17,18], where the collocation polynomial depends on the approximate solution in a fixed number of previous time steps, with the aim of increasing the order of convergence of classical one-step collocation methods, without additional computational cost at each time step, and at the same time obtaining highly stable methods. This idea has been already proposed for the numerical solution of ODEs [19][20][21] (see also [11], Section V.3), and afterward modified in [12], by also using the inherent quadratic technique [22][23][24]. We also underline that they have high uniform order, thus they do not suffer from the order reduction phenomenon, well-known in the ODEs context [9].…”

“…Theorem 3. Letε(t) := y(t) −P(t) be the error of the discretized collocation method (19) and (20) and let p = m + r. Suppose that i.…”

Multivalue Collocation Methods for Ordinary and Fractional Differential Equations

“…, m have been derived by imposing not all the order conditions up to p = 2m + 1, but just requiring the fulfillment of the first m + s order conditions in (10). This procedure corresponds to relaxing some of the interpolation/collocation conditions in (11) and (12), and the corresponding methods are called two-step almost collocation methods.…”

Multivalue Almost Collocation Methods with Diagonal Coefficient Matrix

“…Collocation is a technique which approximates the solution with continuous approximants belonging to a finite dimensional space (usually algebraic polynomials). The approximation satisfies interpolation conditions at the grid points and satisfies the differential equations on the collocation points [ 46 , 48 , 49 , 51 , 52 , 56 ]. Those methods are very effective because they permit to avoid the order reduction typical of Runge Kutta methods, also in presence of stiffness.…”