DEVELOPMENT OF BLOCK UNIFICATION HYBRID LINEAR MULTISTEP METHODS FOR FLUID FLOW EQUATIONS

Abstract

Many physical problems are modelled as differential equations which are either ordinary or partial. These equations require solutions that can be obtained analytically or by the use of numerical methods. For differential equations of higher order, it is almost impossible to obtain solutions analytically, thus the necessity for numerical techniques/methods. This difficulty is the motivation for this study. The study focus on formulation and development of block unification linear multi-step method for the numerical solution of fluid flow equations. with application to both initial and boundary value problems. For this purpose, a Chebyshev polynomials valid in interval [-1,1] and with respect to weight function   2 1 1 x xw   was employed as basis function for the development of continuous hybrid schemes in a collocation and interpolation technique. In order to make the continuous methods self-starting, some block methods of discrete hybrid form were derived. The methods were analysed using appropriate existing theorems to investigate their consistency, zero-stability, convergence and the investigation shows that the developed methods were consistent, zero-stable and hence convergent. These methods were of order three, four and five; with minimal error constants of 001389 .02, 0006944.03,001541.0 4    p CpCpC respectively. The methods were implemented on fifteen (15) test problem from the literature to show the accuracy, efficiency and effectiveness of the methods. It is observed that the proposed methods have maximum error of 6.980 x 10-28 for the oscillatory problem from ship dynamics compared with maximum error of 2.846 x 10-7 obtained from PredictorCorrector method found in the literature. Also for the purpose of comparison, it was observed that the results obtained from the developed methods were validated with Runge- Kutta method and some results in the existing methods which shows an excellent agreement.