Abstract
The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods for fractional differential equations. The construction of the methods together with the convergence and stability analysis are reported and some numerical experiments are carried out to show the efficiency of the proposed methods.
Highlights
Numerical collocation is an effective technique for the approximation of solutions to a given functional equation by means of a continuous approximate belonging to a finite dimensional space spanned by functions chosen in accordance with the qualitative behavior of the exact solution
We focus on spline collocation methods, which were first introduced by Blank [52], the main contribution to the development and analysis of these methods has been given in [19,29,30,53]
We observe that the Runge–Kutta method exhibits order reduction, while this is not the case for multivalue collocation and almost collocation methods, having order 4 but uniform order 2, it suffers from order reduction on stiff problems, as is visible from
Summary
Numerical collocation is an effective technique for the approximation of solutions to a given functional equation by means of a continuous approximate belonging to a finite dimensional space spanned by functions chosen in accordance with the qualitative behavior of the exact solution. The case of stiff differential problems [1,2,3,37,38], commonly arising from spatially discretized time-dependent partial differential equations This problem commonly exposes numerical schemes to the order reduction phenomena, typically characterizing low stage-order methods such as Runge–Kutta methods on Gaussian collocation points [1]. These methods are free from order reduction, as it happens for classical collocation methods We will describe in the two subsections two different choices which lead to two-step collocation methods and Nordsieck GLM collocation methods
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