New 4(3) pair two derivative Runge-Kutta method with FSAL property for solving first order initial value problems

Abstract

A new embedded Two Derivative Runge-Kutta method (TDRK) based on First Same As Last (FSAL) technique for the numerical solution of first order Initial Value Problems (IVPs) is derived. We present an embedded 4(3) pair explicit fourth order TDRK method with a 'small' principal local truncation error coefficient. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of our method in comparison with other existing embedded Runge-Kutta methods (RK) of the same order.

A new embedded 4(3) pair of modified two-derivative Runge–Kutta methods with FSAL property for the numerical solution of the Schrödinger equation

“…It is observed that when v → 0, {γ 1 ,γ 2 ,b 1 ,b 2 } → {1, 1, 103 600 , 97 300 }, then this new modified TDRK method (c, α, β, A,γ ,b) reduces to the third order TDRK method studied by Ahmad et al [39].…”

“…in which the values of c 2 = 1 2 , a 21 = 1 8 are taken from [38], and the values ofb 3 = 1 200 is taken from [39].…”

Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems

“…Problems of such form occur frequently in the scientific areas such as molecular dynamics, quantum mechanics, chemistry, nuclear physics, and electronics. Due to its applications, many researchers are motivated to study the numerical solution of Equation (1) (see [1][2][3][4][5][6][7]). Senu [8] proposed an embedded explicit RKN method for solving oscillatory problems, Fawzi et al [9] derived an embedded 6(5) pair of explicit Runge-Kutta methods for periodic ivps, Franco [10] developed two new embedded pairs of explicit Runge-Kutta methods adapted to the numerical solution of oscillatory problems, and Anastassi [11] constructed a 6(4) optimized embedded Runge-Kutta-Nyström pair for the numerical solution of periodic problems.…”

Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

“…An advantage of these methods is that the number of algebraic order conditions is significantly reduced in comparison with the classical Runge-Kutta methods of the same order, thereby allowing the construction of high-order schemes with only a few stages. Following this, implicit TDRK methods were derived in [13,4], general linear TDRK methods were presented in [1,2,3,11], and recently TDRK methods with optimal phase properties were constructed in [32]. In the case if a good estimate of frequency is known in advance, one can further improve the numerical solution of these methods by incorporating the exponential fitting idea [22,8,29,15] in order to integrate exactly the system who solutions are linear combinations of e ˘iωx (ω is a prescribed frequency).…”

“…In particular, with s " 2 stages we obtain superconvergent schemes of order p " 2s and p " 2s `1, and with s " 3 stages, a method of order p " 2s is derived. This is a significant improvement since previous EFTDRK methods (either explicit or fully implicit) [12,32,4,13] only achieve orders p " 2s for s " 2 stages and p " 2s ´1 for s " 3 stages.…”