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World Journal of Engineering
Research and Technology

( An ISO 9001:2015 Certified International Journal )

An International Peer Reviewed Journal for Engineering Research and Technology

ISSN 2454-695X

Impact Factor : 5.924

ICV : 79.45

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Abstract

SOLUTION OF TWO DIMENSIONAL BURGERS EQUATION BY USING HYBRID CRANK-NICHOLSON AND LAX-FREDRICHS’ FINITE DIFFERENCE SCHEMES ARISING FROM OPERATOR SPLITTING

*J.K. Rotich, J.K. Bitok and M.Z. Mapelu

ABSTRACT

Solving Burgers equation continues to be a challenging problem. Burgers’ equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas of applied mathematics, such as modeling of gas dynamics and traffic flow. It relates to the Navier-Stokes equation for incompressible flow with the pressure term removed. So far the methods that have been used to solve such equations are: Alternative Direction Implicit (ADI) methods, Variation of Iteration Method (VIM), locally one dimensional method and Finite Difference Method (FDM) approach which is used in this work. In this paper the pure Crank-Nicholson (CN) Scheme and Crank-Nicholson-Lax-Fredrichs’ (CN-LF) method is developed by Operator Splitting. Crank-Nicholson-Du-Fort and Frankel is an hybrid scheme made by combining the Crank-Nicholson and Lax-Fredrich schemes. Lax-Friedrichs’ scheme is conditionally stable and an explicit scheme. The developed schemes are solved numerically using initially solved solution via Hopf-Cole transformation and separation of variables to generate the initial and boundary conditions. Analysis of the resulting schemes was found to be unconditionally stable. The results of the hybrid scheme are found to compare well with those of the pure Crank-Nicholson.

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