A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

F F Ngwane,S N Jator

doi:10.1155/2017/9293530

Abstract

In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

Highlights

In what follows, we consider the numerical solution of the general second-order initial value problems (IVPs) of the form y󸀠󸀠 = f (x, y, y󸀠), y (x0) = y0, (1)y󸀠 (x0) = y0󸀠, x ∈ [x0, xN], where f : R × R2m → R2m, N > 0 is an integer, and m is the dimension of the system
We consider the numerical solution of the general second-order IVPs of the form y󸀠󸀠 = f (x, y, y󸀠), y (x0) = y0, (1)
Our objective is to present a block hybrid trigonometrically fitted Runge-Kutta-Nystrom method (BHTRKNM) that is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictorcorrector methods

Summary

Introduction

Fewer methods have been proposed for directly solving second-order IVPs in which the first derivative appears explicitly (see Vigo-Aguiar and Ramos [11], Awoyemi [12], Chawla and Sharma [13], Mahmoud and Osman [14], Franco [15], and Jator [16, 17]). Ramos and Vigo-Aguiar [24], Franco and Gomez [25], and Ozawa [26]) Most of these methods are restricted to solving special second-order IVPs in a predictor-corrector mode.

Development of the BHTRKNM

Error Analysis and Stability

Numerical Examples

Hyperbolic PDE

Conclusion