Natural Volterra Runge-Kutta methods

Abstract

A very general class of Runge-Kutta methods for Volterra integral equations of the second kind is analyzed. Order and stage order conditions are derived for methods of order p and stage order q = p up to the order four. We also investigate stability properties of these methods with respect to the basic and the convolution test equations. The systematic search for A- and V0-stable methods is described and examples of highly stable methods are presented up to the order p = 4 and stage order q = 4

User-Friendly Expressions of the Coefficients of Some Exponentially Fitted Methods

“…Further developments of this research will be oriented to the reformulation, through C and S functions, of existing methods for ordinary differential equations [ 2 , 17 , 20 , 25 , 26 , 28 , 37 – 39 , 41 , 42 , 44 , 48 , 51 , 53 , 56 , 77 , 80 ], integral equations [ 5 – 8 , 10 , 11 , 24 , 29 , 32 , 34 , 55 , 71 ], stochastic problems [ 9 , 12 , 13 , 18 , 19 , 29 , 47 ], fractional equations [ 12 , 13 , 21 , 22 , 36 ], partial differential equations [ 57 – 59 , 66 , 74 ].…”

High-order and V_0-stable methods for solving the Volterra integral equations of the second kind

“…The VIEs are widely applied in many applications in Physics, Economics, Engineering, Biology [5,14,19,25]. In the past five decades, one of the main challenges has been to develop some efficient approximate methods for the solution of these types of equations [1,2,3,7,10,11,12,13,15,16,17,20,21,22]. The stability analysis of these methods is mostly focused on A-stability property by studying the behavior of the following test equation…”

“…In [17], they also investigated the numerical stability of a class of Runge-Kutta methods for the solution of the VIEs (1), and introduced some V 0 -stable methods of orders p = 3, 4 with q = 4, 8 stages, respectively. Conte et al in [12] introduced a general class of Runge-Kutta methods for VIEs of the second kind. They also investigated the A-and V 0 -stability of their methods with respect to the basic and the convolution test equations.…”

Construction of efficient general linear methods for stiff Volterra integral equations

“…To solve stiff VIEs numerically, the applied method must have some reasonably wide region of absolute stability [6,8,9,13]. In this regard, A-and V 0 -stable RungeKutta methods for VIEs have been introduced in [1,5,11]. Like general linear methods (GLMs) which have been introduced as a unifying framework for the traditional methods for solving initial value problems [3,12], Izzo et al [10] investigated the class of GLMs of order p and stage order q = p for the numerical solution of (1) to analyze stability properties of the method conveniently.…”