General Nyström methods in Nordsieck form: Error analysis

Abstract

The paper is concerned with the analysis of the error associated to a family of multi-value numerical methods for the solution of initial value problems based on special second order ordinary differential equations. Such methods, denoted as General Nyström methods, provide at each step point an approximation to the Nordsieck vector associated to the solution of the problem. Order issues for such methods based on the theory of rooted trees are here provided, as well as an accuracy analysis is carried out, leading to a representation of the local truncation error

Multivalue Almost Collocation Methods with Diagonal Coefficient Matrix

“…Thanks to the structure of the coefficient matrix, those methods can be easily parallelized, so the computational effort can be reduced. In the future we aim to construct such types of methods for different operators such as stochastic differential equations [ 15 , 21 , 31 ], fractional differential equations [ 2 , 10 , 13 , 16 ], partial differential equations [ 1 , 11 , 14 , 20 , 30 , 32 , 35 , 38 , 40 ], Volterra integral equations [ 8 , 9 , 12 , 17 , 23 ], second order problems [ 26 , 37 ], oscillatory problems [ 19 , 22 , 24 , 33 , 36 , 53 ], as well as to the development of algebraically stable high order collocation based multivalue methods [ 18 , 29 ].…”

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

“…Further issues of this research will focus on oscillatory problems [46,47] and in particular on the application of multistep collocation methods to periodic integral equations [48,49]. Moreover, it seems reasonable to consider the possibility of employing collocation spaces based on functions other than polynomials, as in [50][51][52] and similarly as in the case of oscillatory problems [53], and merge into the numerical scheme as many known qualitative properties of the continuous problem as possible, in a structure-preserving perspective [54].…”

“…We impose that at the collocation pointsP(t) satisfies the VIDE (39), where the integrals appearing in both the lag term (53) and the increment function (54) are approximated by the quadrature formulae defined in (55), and we setỹ n+1 =P(t n + h). Thus the discretized multistep method is…”

“…i.e., they are obtained by applying the quadrature Formula (55) to the integrals appearing in (53) and (54).…”

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

“…The structure-preserving approach to SDEs will next be devoted to stochastic Hamiltonian problems [7,8] and the stochastic extension of existing deterministic approaches [32][33][34]. A stochastic version of the non-polynomial fitting for oscillatory problems will also be addressed [35][36][37][38].…”