AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION

Murat Sarı; Huseyin Tunc

AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION

Abstract

This article has explored a hybrid numerical approach in analysis of the Burgers equation with involving steep gradients. The technique is based on a quadratic B-spline finite element method in strong form for space variation. This paper discovers how to find an α-family optimization approach for temporal variations. The proposed method has been shown to be unconditionally stable for α≥0.5. Yet, the efficiency of the proposed scheme on relatively coarse grids has been demonstrated. The numerical illustrations show that the present method has been seen to be more accurate than the literature and effectively captures the shock behaviours.

Keywords

References

[1] Sari M., Gurarslan G. (2009), A sixth-order compact finite difference scheme to the numerical solutions of Burgers’ equation. Applied Mathematics and Computation, 208, 475-483.
[2] Wang J., Warnecke G. (2003), Existence and uniqueness of solutions for a non-uniformly parabolic equation. Journal of Differential Equations, 189, 1–16.
[3] Miller EL. (1966), Predictor–corrector studies of Burger’s model of turbulent flow, M.S. Thesis University of Delaware, Newark, Delaware.
[4] Kutluay S., Bahadir A.R., Ozdes A. (1999), Numerical solution of one-dimensional Burgers’ equation: explicit and exact-explicit finite difference methods. Journal of Computational and Applied Mathematics, 103, 251–261.
[5] Dag I., Irk D. Saka B. (2005), A numerical solution of the Burgers’ equation using cubic B-splines. Applied Mathematics and Computation, 163, 199–211.
[6] Bahadir AR., Saglam M. (2005), A mixed finite difference and boundary element approach to one-dimensional Burgers’ equation. Applied Mathematics and Computation, 160, 663–673.
[7] Ramadan MA., El-Danaf TS. (2005), Numerical treatment for the modified Burgers equation, Mathematics and Computers in Simulation , 70, 90–98.
[8] Kutluay S., Esen A., Dag I. (2004), Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J. Computational and Applied Mathematics, 167, 21-33.

[9] Jiwari R. (2015), A hybrid numerical scheme for the numerical solution of the Burgers’ equation. Computer Physics Communications, 188, 50-67.
[10] Raslan KR. (2003), A collocation solution for Burgers equation using quadratic B-spline finite elements. International Journal of Computational and Applied Mathematics, 80, 931-938.
[11] Talwar J., Mohanty RK., (2016), Singh S. A new algorithm based on spline in tension approximation for 1D parabolic quasi-linear equations on a variable mesh. International Journal of Computational and Applied Mathematics, 93, 1771-1786.
[12] Bahadir AR., Saglam M. (2005), A mixed finite difference and boundary element approach to one-dimensional Burgers’ equation. Applied Mathematics and Computation, 160, 663–673.
[13] Liao W., Zhu J. (2011), Efficient and accurate finite difference schemes for solving one-dimensional Burgers’ equation. International Journal of Computational and Applied Mathematics, 88, 2575-2590.
[14] Kutluay S., Esen A. (2004), A lumped Galerkin method for solving the Burgers equation. International Journal of Computational and Applied Mathematics, 81, 1433-1444.
[15] Verma AK., Verma L. (2004), Higher order time integration formula with application on Burgers’ equation. International Journal of Computational and Applied Mathematics, 92, 756-771.
[16] Shao L., Feng X., He Y. (2011), The local discontinuous Galerkin finite element method for Burgers equation. Mathematical and Computer Modelling, 54, 2943-2954.
[17] Wang W., Lu T. (2005), The alternating segment difference scheme for Burgers’ equation. International Journal for Numerical Methods in Fluids, 49, 1347–1358.
[18] Kannan K., Wang Z.J. (2012), A high order spectral volume solution to the Burgers’ equation using the Hopf–Cole transformation. International Journal for Numerical Methods in Fluids, (69), 781–801.
[19] Chen J., Chen Z., Cheng S., Zhan J. (2015), Multilevel Augmentation Methods for Solving the Burgers’ Equation. Numerical Methods for Partial Differential Equations, 31, 1665–1691.
[20] Wu Y., Wu X. (2004), Linearized and rational approximation method for solving non-linear Burgers’ equation. International Journal for Numerical Methods in Fluids; 45, 509–525.
[21] Hopf E. (1950), The partial differential equation u_t+uu_x=εu_xx . Communucations on Pure and Applied Mathematics, 9, 201-230.
[22] Cole JD. (1951), On a quasi-linear parabolic equation in aerodynamics. Quarterly of Applied Mathematics, 9, 225-236.
[23] Aksan EN. (2006), Quadratic B-spline finite element method for numerical solution of the Burgers equation. Applied Mathematics and Computation, 174, 884-896.
[24] Dag I., Saka B., Boz A. B-spline Galerkin methods for numerical solutions of Burgers’ equation. Applied Mathematics and Computation 2005; 166:506-522.
[25] Prenter PM. Splines and Variational Methods, John Wiley & Sons, New York, 1975.
[26] Reddy JD. An Introduction to the Finite Element Method, 3rd edition, McGraw-Hill, Singapore, 2006. [27] Tsai C., Shih Y., Lin Y., Wang H. Tailored finite point method for solving one-dimensional Burgers’ equation. International Journal of Computational and Applied Mathematics DOI: 10.1080/00207160.2016.1148812
[28] Kutluay S., Bahadir AR., Ozdes A. Numerical solution of one-dimensional Burgers’ equation: Explicit and exact-explicit finite difference methods. Journal of Computational and Applied Mathematics 1999; 103:251–261.
[29] Jiwari R., Mittal R.C., Sharma K.K. A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation. Applied Mathematics and Computation 2013; 219:6880-6891.
[30] Mittal R.C., Jiwari R., Sharma K.K. A numerical scheme based on differential quadrature method to solve time dependent Burgers' equation. Engineering Computations 2013; 30:117-131.
[31] Mittal R.C., Jiwari R. A numerical scheme based on differential quadrature method to solve time dependent Burgers' equation. International Journal of Numerical Methods for Heat and Fluid Flow. 2012; 22: 880-895.

Details

Primary Language

English

Subjects

Engineering

Journal Section

Research Article

Authors

Murat Sarı This is me
0000-0003-0508-2917
Türkiye

Huseyin Tunc This is me
0000-0001-6450-5380
Türkiye

Publication Date

September 1, 2017

Submission Date

November 19, 2016

Acceptance Date

July 15, 2017

Published in Issue

Year 2017 Volume: 35 Number: 3

IZ

https://izlik.org/JA52GP82KC

Cite

RIS / Bibtex

APA

Sarı, M., & Tunc, H. (2017). AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION. Sigma Journal of Engineering and Natural Sciences, 35(3), 369-386. https://izlik.org/JA52GP82KC

AMA

1.Sarı M, Tunc H. AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION. SIGMA. 2017;35(3):369-386. https://izlik.org/JA52GP82KC

Chicago

Sarı, Murat, and Huseyin Tunc. 2017. “AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION”. Sigma Journal of Engineering and Natural Sciences 35 (3): 369-86. https://izlik.org/JA52GP82KC.

EndNote

Sarı M, Tunc H (September 1, 2017) AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION. Sigma Journal of Engineering and Natural Sciences 35 3 369–386.

IEEE

[1]M. Sarı and H. Tunc, “AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION”, SIGMA, vol. 35, no. 3, pp. 369–386, Sept. 2017, [Online]. Available: https://izlik.org/JA52GP82KC

ISNAD

Sarı, Murat - Tunc, Huseyin. “AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION”. Sigma Journal of Engineering and Natural Sciences 35/3 (September 1, 2017): 369-386. https://izlik.org/JA52GP82KC.

JAMA

1.Sarı M, Tunc H. AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION. SIGMA. 2017;35:369–386.

MLA

Sarı, Murat, and Huseyin Tunc. “AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION”. Sigma Journal of Engineering and Natural Sciences, vol. 35, no. 3, Sept. 2017, pp. 369-86, https://izlik.org/JA52GP82KC.

Vancouver

1.Murat Sarı, Huseyin Tunc. AN OPTIMIZATION TECHNIQUE IN ANALYZING THE BURGERS EQUATION. SIGMA [Internet]. 2017 Sep. 1;35(3):369-86. Available from: https://izlik.org/JA52GP82KC