FINITE DIFFERENCE SCHEMES ARISING FROM OPERATOR SPLITTING FOR SOLVING TWO DIMENSIONAL SYSTEM OF BURGERS’ EQUATION

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 fluid 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) which is used in this work. The study developed the pure Crank-Nicholson (CN), Crank-Nicholson-Du-Fort and Frankel (CN-DF), Crank-Nicholson- LaxFriedrichs‟(CN-LF) and Crank-Nicholson- Du-Fort and Frankel-Lax-Friedrichs‟ (CN-DF-LF) schemes by Operator Splitting. Crank-Nicholson-Du-Fort and Frankel is an hybrid scheme made by combining the Crank-Nicholson and Du-Fort and Frankel schemes which are both unconditionally stable but the Du-fort scheme is explicit while the Crank-Nicholson scheme is implicit and the Crank-Nicholson- Lax-Friedrichs‟ scheme is a hybrid scheme made up of combining the CrankNicholson and Lax-Friedrichs‟ scheme. Lax-Friedrichs‟ scheme is conditionally stable and an explicit scheme while the Crank-Nicholson- Du-Fort and Frankel-LaxFriedrichs‟ method is a hybrid scheme made by combining the Crank-Nicholson, Du-Fort and Frankel and Lax-Friedrichs‟ schemes. Crank-Nicholson-Du-Fort and Frankel is an hybrid scheme made by combining the Crank-Nicholson and Du-Fort and Frankel schemes which are both unconditionally stable but the Du-fort scheme is explicit while the Crank-Nicholson scheme is implicit. The developed schemes were solved numerically using MATLAB was used to generate the results. Analysis of the schemes showed that they are consistent, convergent and stable.