Construction of Efficient General Linear Methods for Non-Stiff Differential Systems

Abstract

This paper describes the construction of explicit general linear methods in Nordsieck form with inherent quadratic stability and large areas of the stability region. After satisfying order and inherent quadratic stability conditions, the remaining free parameters are used to find the methods with large area of region of absolute stability. Examples of methods with p = q + 1 = s = r and p = q = s = r − 1 up to order 6 are given.

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].…”

“…and t n0 = t n . We observe as the method (22) requires, at each step, the solution of a nonlinear system of (m + 1)d equations in the stage values Y Theorem 5. Assume that the kernel k(t, η, y) and the function g(t) in (1) are sufficiently smooth.…”

“…Assume that the kernel k(t, η, y) and the function g(t) in (1) are sufficiently smooth. Then the method (21) and (22) has uniform order p, i.e.,…”

Explicit General Linear Methods With a Large Stability Region for Volterra Integro-Differential Equations

“…Considering this point, in Section 3, we construct explicit GLMG schemes based on explicit DIMSIMs for ODEs with and without RKS property in which the free parameters of the methods are used to have RKS property for DIMSIMs or to extend the absolute stability region of DIMSIMs with the aim of maximizing the its area. The used strategy to achieve the maximum value for area is the same as that in [2,3,5,6,23,24,25,28]. Verification of the theoretical results, efficiency, and capability of the constructed methods in solving nonstiff and mildly stiff VIDEs together with comparison with implicit GLMG are given in Section 4.…”

Two-step collocation methods for fractional differential equations

“…Our purpose is to obtain higher order collocation methods. A successful strategy to increase the order of convergence consists of applying multivalue methods, as done in the case of ordinary differential equations [13,28,29], where the multistep collocation methods were modified in [13] by also using the inherent quadratic stability technique [3,4]. Multivalue methods were also adopted in Volterra integral equations [11] and Volterra integro-differential equations [8].…”