Ordinary Hybrid Finite Difference Methods for Solving Burgers’ Equation.
Keywords:
Burgers’ Equation, Crank-Nicholson; Lax-Friedrichs’; Du Fort and Frankel methods.Abstract
Burgers’ equation appears as a model in turbulence and gas dynamics. We construct hybrid finite difference schemes from ordinary finite difference methods for solving this equation. Among the hybrid methods developed are the Crank-Nicholson-Du Fort and Frankel and Crank-Nicholson- Lax-Friendrich’s and Du Fort and Frankel. We determine that the Du Fort and Frankel discretization have an improvement effect on other finite difference schemes whereas the Lax- Friedrich’s method reduces their efficacy. We note that the Du Fort and Frankel method increases the number of grid points involved by one. The increase of the grid points is responsible for the improved accuracy of the Crank-Nicholson and the Hybrid Crank-Nicholson-Lax-Friedrich’s, methods. The hybrid Crank-Nicholson-Lax-Friedrich’s,-Du Fort and Frankel scheme is the most accurate.
References
Ames, W.F. (1977). Numerical Methods for Partial Differential Equations in Engineering. Academic Press, New York.
Chapra S.C and Canale R.P (1998). Numerical methods for Engineers WCB/ McGraw-hill.
Drazin P.G. and Johnson R.S. (1996).Solitons on Introduction. Cambridge University press, 1996.
Jain M.K (1984). Numerical Solution of Differential equations (2nd edition). Wiley Eastern Limited..
Mitchel A. R. and Griffiths D. F. (1980). The Finite Difference Methods in Partial Differential Equations. John Wiley and Sons.
Morton K. W. and Mayers D. F. (1994).Numerical Solution of Partial Differential Equations Cambridge University Press.
Rahman M. (1994). Partial Differential Equations. Computational Mechanics Publications, Southamptom Boston.
Rao K.S. (2005).Numerical Methods for Scientists and Engineers (2nd Edition). Prentice hall of India, Private Limited, New Delhi.
Wood W.L(2006). An exact solution for Burgers equation. Commun. Numer. Meth Engng 2006, 22 797-798.