### A hybrid approach for the regularized long wave-Burgers equation

#### Abstract

In this paper, a new hybrid approach based on sixth-order finite difference and seventh-order weighted essentially non-oscillatory finite difference scheme is proposed to capture numerical simulation of the regularized long wave-Burgers equation which represents a balance relation among dissipation, dispersion and nonlinearity. The corresponding approach is implemented to the spatial derivatives and then MacCormack method is used for the resulting system. Some test problems discussed by different researchers are considered to apply the suggested method. The produced results are compared with some earlier studies, and to validate the accuracy and efficiency of the method, some error norms are computed. The obtained solutions are in good agreement with the literature. Furthermore, the accuracy of the method is higher than some previous works when some error norms are taken into consideration.

#### Keywords

#### Full Text:

PDF#### References

Bona, J.L., Pritchard, W.G. & Scott, L.R. (1981). An evaluation of a model equation for water waves. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 302(1471), 457-510.

Zhao, X., Li, D. & Shi, D. (2008). A finite difference scheme for RLW-Burgers equation. Journal of Applied Mathematics & Informatics, 26(3,4), 573-581.

Al-Khaled, K., Momani, S. & Alawneh, A. (2005). Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations. Applied Mathematics and Computation, 171(1), 281-292.

Alquran, M. & Al-Khaled, K. (2011). Sinc and solitary wave solutions to the generalized Benjamin-Bona-Mahony-Burgers equations. Physica Scripta, 83(6), 6 pages.

Arora, G., Mittal, R.C. & Singh, B.K. (2014). Numerical solution of BBM-Burgers equation with quartic B-spline collocation method. Journal of Engineering Science and Technology, 9, 104-116.

Che, H., Pan, X., Zhang, L. & Wang, Y. (2012). Numerical analysis of a linear-implicit average scheme for generalized Benjamin-Bona-Mahony-Burgers equation. Journal of Applied Mathematics, 2012, 14 pages.

Omrani, K. & Ayadi, M. (2008). Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation. Numerical Methods for Partial Differential Equations, 24(1), 239-248.

Zarebnia, M. & Parvaz, R. (2013). Cubic B-spline collocation method for numerical solution of the Benjamin-Bona-Mahony-Burgers equation. WASET International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 7(3), 540-543.

Zarebnia, M. & Parvaz, R. (2016). On the numerical treatment and analysis of Benjamin-Bona-Mahony-Burgers equation. Applied Mathematics and Computation, 284, 79-88.

Shen, Y. & Zha, G. (2008). A Robust seventh-order WENO scheme and its applications, 46th AIAA Aerospace Sciences Meeting and Exhibit, 2008 Jan 7-10, Reno, Nevada, AIAA 2008-075.

Shen, Y. & Zha, G. (2010). Improved seventh order WENO scheme. 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010 Jan 4-7, Orlando, Florida, AIAA 2010-1451 .

Zahran, Y.H. & Babatin, M.M. (2013). Improved ninth order WENO scheme for hyperbolic conservation laws. Applied Mathematics and Computation, 219(15), 8198-8212.

Jiang, G.S. & Shu, C.W. (1996). Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126(1), 202-228.

Wang, Z.J. & Chen, R.F. (2001). Optimized weighted essentially non-oscillatory schemes for linear waves with discontinuity. Journal of Computational Physics, 174(1), 381-404.

Ponziani, D., Prizzoli, S. & Grasso, F. (2003). Develpoment of optimized weighted-ENO schemes for multiscale compressible flows. International Journal for Numerical Methods in Fluids, 42(9), 953-977.

Balsara, D.S. & Shu, C.W. (2000). Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. Journal of Computational Physics, 160(2), 405-452.

Pirozzoli, S. (2002). Conservative hybrid compact-WENO schemes for shock-turbulence interaction. Journal of Computational Physics, 178(1), 81-117.

Kim, D. & Kwon, J.H. (2005). A high-order accurate hybrid scheme using a central flux scheme and a WENO scheme for compressible flowfield analysis. Journal of Computational Physics, 210(2), 554-583.

Shen, Y.Q. & Yang, G.W. (2007). Hybrid finite compact WENO schemes for shock calculation. International Journal for Numerical Methods in Fluids, 53(4), 531-560.

Sari, M., Gürarslan, G. & Zeytinoglu, A. (2010). High-order finite difference schemes for solving the advection diffusion equation. Mathematical and Computational Applications, 15(3), 449-460.

Zeytinoglu, A. (2010). Some approximate solutions of Burgers equations. MSc Thesis. Suleyman Demirel University.

Liu, X.D., Osher, S. & Chan, T. (1994). Weighted essentially non-oscillatory schemes. Journal of Computational Physics, 126, 200-212.

Xie, P., (2007). Uniform weighted compact/non-compact schemes for shock/boundary layer interaction. PhD Thesis. The University of Texas.

Pletcher, R.H., Tannehill, J.C. & Anderson, D.A. (2013). Computational fluid mechanics and fluid transfer. Taylor&Francis.

DOI: http://dx.doi.org/10.11121/ijocta.01.2018.00461

### Refbacks

- There are currently no refbacks.

Copyright (c) 2017 Murat Sari

This work is licensed under a Creative Commons Attribution 4.0 International License.