A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations

Adegoke Stephen Olaniyan,Omolara Fatimah Bakre,Moses Adebowale Akanbi

doi:10.11648/j.pamj.20200905.11

Abstract

In recent times, the use of different types of mean in the derivation of explicit Runge-Kutta methods had been on increase. Researchers have explored explicit Runge-Kutta methods derivation by using different types of mean such as geometric mean, harmonic mean, contra-harmonic mean, heronian mean to name but a few; as against the conventional explicit Runge-Kutta methods which was viewed as arithmetic mean. However, despite efforts to improve the derivation of explicit Runge-Kutta methods with use of other types of mean, none has deemed it fit to extend this notion to implicit Runge-Kutta methods. In this article, we present the use of heronian mean as a basis for the construction of implicit Runge-Kutta method in a way of improving the conventional method which is arithmetic mean based. Numerical results was conducted on ordinary differential equations which was compared with the conventional two-stage fourth order implicit Runge-Kutta (IRK4) method and two-stage third order diagonally implicit Runge-Kutta (DIRK3) method. The results presented confirmed that the new scheme performs better than these numerical methods. A better Qualitative properties using Dalquist test equation were established.

Highlights

Runge-Kutta methods are among the oldest and best understood methods in numerical analysis which have prompted researchers to explore them
Much work have been done recently in the formulation of explicit Runge-Kutta based on averages other than the conventional arithmetic mean for the numerical solution of ordinary differential equation
A new 4th order embedded method based on the harmonic mean was constructed by Yaacob N. and Sanugi B. (1998) [20] in which harmonic mean was embedded in the arithmetic mean viewed Runge-kutta methods

Summary

Introduction

Runge-Kutta methods are among the oldest and best understood methods in numerical analysis which have prompted researchers to explore them. B. [11] developed a new 4th order Runge-Kutta method based on the contra-harmonic mean. They asserted that it is accurate compared to its equivalents. (2011) [17] developed a new Fifth-Order Fifth Stage Runge-Kutta method on heronian mean wherein heronian mean was used in the construction of 5th stage explicit Runge-Kutta methods. A new 4th order embedded method based on the harmonic mean was constructed by Yaacob N. and Sanugi B. The method was derived based on harmonic mean and was confirmed to be better than any third order of any form of explicit Runge-Kutta methods.

Materials and Methods

Convergence of the Scheme

Numerical Experiment

Conclusion