Continuous two-step Runge–Kutta methods for ordinary differential equations

Abstract

New classes of continuous two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are derived. These methods are developed imposing some interpolation and collocation conditions, in order to obtain desirable stability properties such as A-stability and L-stability. Particular structures of the stability polynomial are also investigated.

Construction and implementation of highly stable two-step continuous methods for stiff differential systems

“…Additional results confirming that two-step continuous methods preserve the order of convergence for stiff problems can be found in [12,14,15].…”

“…Two-step continuous methods (1.2) were introduced in [12,13,19,21] and further investigated in [9,14,15], also in the context of Volterra Integral Equations [10,11]. In this paper we follow a similar approach to that in [13], but restrict our attention to methods of order p = m and stage order q = p to avoid order reduction phenomenon [3] for stiff equations.…”

Polarization and Electromagnetic Emissions of Natural Crystalline Structures upon Acoustic Excitation

“…Solutions of Eq. 4 were performed using the Runge -Kutta numerical method [25,26]. To solve the equation, the parameters were chosen with the following values close to the parameters of the ore sample: the resistance of the sample material is R = 1 kΩ, the dielectric constant inside the cavity of the medium ε = 4, and the cross section of the capacitor cavity area S = 4×10 -4 m 2 .…”

Natural Continuous Extension Implicit Runge-Kutta Method for Solving Nonlinear DDE in Blood Cell Production

“…[11] investigated the global attractivity for the model Eq. (1), though Ambrosio and Jackiewicz [2] constructed continuous extensions to Runge-Kutta (RK) methods for ordinary differential equations. Baker and Paul [3] discussed about the practical determination of stability regions when various fixed step-size RK methods, joined with continuous extensions, are applied to the linear delay differential equation with fixed delay.…”