A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value Problems

F F Ngwane,S N Jator

doi:10.1155/2015/343295

Abstract

A family of Enright’s second derivative formulas with trigonometric basis functions is derived using multistep collocation method. The continuous schemes obtained are used to generate complementary methods. The stability properties of the methods are discussed. The methods which can be applied in predictor-corrector form are implemented in block form as simultaneous numerical integrators over nonoverlapping intervals. Numerical results obtained using the proposed block form reveal that the new methods are efficient and highly competitive with existing methods in the literature.

Highlights

Many real life processes in areas such as chemical kinetics, biological sciences, circuit theory, economics, and reactions in physical systems can be transformed into systems of ordinary differential equations (ODE) which are generally formulated as initial value problems (IVPs)
Second derivative methods with polynomial basis functions were proposed to overcome the Dahlquist [8] barrier theorem whereby the conventional linear multistep method was modified by incorporating the second derivative term in the derivation process in order to increase the order of the method, while preserving good stability properties
We consider some standard problems: stiff, oscillatory, linear, and nonlinear systems that appear in the literature to experimentally illustrate the accuracy and efficiency of the Enright’s second derivative methods (ESDM) (10) which is implemented in block form

Summary

Introduction

Many real life processes in areas such as chemical kinetics, biological sciences, circuit theory, economics, and reactions in physical systems can be transformed into systems of ordinary differential equations (ODE) which are generally formulated as initial value problems (IVPs). Different methods including the Backward Differentiation Formula (BDF) have been used to solve stiff systems. Many classical numerical methods including RungeKutta methods, higher derivative multistep schemes, and block methods have been constructed for solving oscillatory initial value problems (see Butcher [11, 12], Brugnano and Trigiante [13, 14], Ozawa [15], Nguyen et al [16], Berghe and van Daele [17], Vigo-Aguiar and Ramos [18], and Calvo et al [19]). Obrechkoff [20] proposed a general multiderivative method for solving systems of ordinary differential equations. We propose a numerical integration formula which more effectively copes with stiff and/or oscillatory.

Derivation of the Family of Methods

Error Analysis and Stability

Stability

Numerical Illustration

Conclusion

Conflict of Interests