𝑃-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems

Abstract

Abstract. In this paper /'-stable methods of 0(hb) and 0(/i8) with minimal phase-lag (frequency distortion) are derived. Numerical results for both linear and nonlinear problems are presented.

Numerical simulation of second-order initial-value problems using a new class of variable coefficients and two-step semi-hybrid methods

“…In what follows, we will mention some references. The Runge–Kutta methods, 1–10 multistep methods, 11–17 and Runge–Kutta–Nyström methods 18–42 are some of the approaches that can be used for solving a second-order differential equation. Since the Störmer–Cowell multistep procedure with more than two steps suffers from orbital instability issues, some improved versions of the Störmer–Cowell method were proposed by Gautschi 8 and Stiefel and Bettis.…”

A New Explicit Four-Step Symmetric Method for Solving Schrödinger’s Equation

“…The first category involves methods with constant coefficients, and the second one involves methods using variable coefficients. Methods of the first type are developed in Gautschi [2], Jain and et al [3], and Steifel and Bettis in [4], while methods of the second kind are presented in Chawla and et al [5,6], Dahlquist [7], Franco [8], Lambert and Watson [9], Krishnaiah [10], Simos and et al [11,12], Saldanha and Achar [13], Achar [14], and Daele and Vanden Berghe [15].…”

A new efficient implicit four-step method with vanished phase-lag and some of its derivatives for the numerical solution of the radial Schr¨odinger equation

“…Large research on the algorithmic development of numerical methods for the solution of the Schrödinger equation has been done in the last decades. The aim and scope of this research is the construction of fast and reliable algorithms for the solution of the Schrödinger equation and related problems (see for example [2,3,4,5]). Mathematical models in theoretical physics and chemistry, material sciences, quantum mechanics and quantum chemistry, electronics etc.…”

“…Computational methods involving a parameter proposed by Gautschi [12], Jain et al [14] and Steifel and Bettis [30] yield numerical solution of problems of class (1). Chawla and et al [7,8], Anantha krishnaiah [3], Shokri and et al [20,21,22,23,24], Dahlquist [9], Franco [10], Lambert and Watson [15], Simos and et al [25,26,27], Saldanha and Achar [19], Achar [1], and Daele and Vanden Berghe [31] have developed methods to solve problems of class (2). We have organized the paper as follows: In Section 2, we present the preliminary concepts that requisite for theory of the new methodology.…”