Document Type : Article

**Authors**

Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Istanbul 34220, Turkey

**Abstract**

In this study, the Burgers equation is analyzed in both numerically and mathematically by considering various finite element based techniques including Galerkin, Taylor-Galerkin and collocation methods for spatial variation of the equation. The obtained time dependent ordinary differential equation system is approximately solved by α-family of time approximation. All these methods are theoretically explained using cubic B-spline basis and weight functions for a strong form of the model equation. Von Neumann matrix stability analysis is performed for each of these methods and stability criteria are determined in terms of the problem parameters. Some challenging examples of the Burgers equation are numerically solved and compared with the literature and exact solutions. Also, the proposed techniques have been compared with each other in terms of their advantageous and disadvantageous depending on the problem types. The more advantageous method of the three, comparison to other two, has been found out for the special cases of the present problem in detail.

**Keywords**

References:

1. Sari, M. and Gurarslan, G. "A sixth-order compact finite difference scheme to the numerical solutions of Burgers' equation", Appl Math Comput., 208, pp. 475- 483 (2009).

2. Wang, J. and Warnecke, G. "Existence and uniqueness of solutions for a non-uniformly parabolic equation", J. Differ. Equations, 189, pp. 1-16 (2003).

3. Miller, E.L. "Predictor-corrector studies of Burger's model of turbulent flow", MS Thesis, University of Delaware, Newark, Delaware (1966).

4. Kutluay, S., Bahadir, A.R., and Ozdes, A. "Numerical solution of one-dimensional Burgers' equation: explicit and exact-explicit finite difference methods", J. Comput. Appl. Math., 103, pp. 251-261 (1999).

5. Seydaoglu, M. "An accurate approximation algorithm for Burgers' equation in the presence of small viscosity", J. Comput. Appl. Math., 344, pp. 473-481 (2018).

6. Bahadir, A.R. and Saglam, M. "A mixed finite difference and boundary element approach to onedimensional Burgers' equation", Appl. Math. Comput., 160, pp. 663-673 (2005).

7. Sari, M., Tunc, H., and Seydaoglu M. "Higher order splitting approaches in analysis of the Burgers equation", Kuwait J. Sci., 46(1), pp. 1-14 (2019).

8. Hopf, E. "The partial differential equation", Commun. Pur. Appl. Math., 9, pp. 201-230 (1950).

9. Cole, J.D. "On a quasi-linear parabolic equation in aerodynamics", Q. Appl. Math., 9, pp. 225-236 (1951).

10. Talwar, J., Mohanty, R.K., and Singh, S. "A new algorithm based on spline in tension approximation for 1D parabolic quasi-linear equations on a variable mesh", Int. J. Comput. Math., 93, pp. 1771-1786 (2016).

11. Jiwari, R. "A hybrid numerical scheme for the numerical solution of the Burgers' equation", Comput. Phys. Commun., 188, pp. 50-67 (2015).

12. Seydaoglu, M., Erdogan, U., and Ozis, T. "Numerical solution of Burgers' equation with higher order splitting methods", J. Comput. Appl. Math., 291, pp. 410-421 (2016).

13. Korkmaz, A. and Dag, I. "Polynomial based differential quadrature method for numerical solution of nonlinear Burgers' equation", J. Franklin I., 248, pp. 2863-2875 (2016).

14. Bahadir, A.R. and Saglam, M. "A mixed finite difference and boundary element approach to onedimensional Burgers' equation", Appl. Math. Comput., 160, pp. 663-673 (2005).

15. Egidi, N., Maponi, P., and Quadrini, M. "An integral equation method for the numerical solution of the Burgers equation", Comput. Math. Appl., 76(1), pp. 35-44 (2018).

16. Inan, B. and Bahadir, A.R. "An explicit exponential finite difference method for the Burgers' equation", European International Journal of Science and Technology, 2(10), pp. 61-72 (2013).

17. Zeytinoglu, A., Sari, M., and Pasaoglu, B.A. "Numerical simulations of shock wave propagating by a hybrid approximation based on high-order finite difference schemes", Acta Phys. Pol. A, 133, pp. 369-386 (2017).

18. Verma, A.K. and Verma, L. "Higher order time integration formula with application on Burgers' equation", Int. J. Comput. Math., 92, pp. 756-771 (2015).

19. Rouzegar, J. and Sharifpoor, R.A. "A finite element formulation for bending analysis of isotropic and orthotropic plates based on two-variable refined plate theory", Sci. Iran., 22(1), pp. 196-207 (2015).

20. Ak, T., Karakoc, S.B.G., and Biswas, A. "Application of Petrov-Galerkin finite element method to shallow water waves model: modified Korteweg-de Vries equation", Sci. Iran., 24(3), pp. 1148-1159 (2017).

21. Zendehbudi, G. "Combination of projection and Galerkin finite element methods to solve the problem of free convection in enclosures with complex geometries", Sci. Iran, 25(3), pp. 1189-1196 (2018).

22. Kutluay, S., Esen, A., and Dag, I. "Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method", J. Comput. Appl. Math., 167, pp. 21-33 (2004).

23. Sari, M. and Tunc, H. "An optimization technique in analyzing the Burgers equation", Sigma Journal of Engineering and Natural Sciences, 35(3), pp. 369-386 (2017).

24. Kutluay, S. and Esen, A. "A lumped Galerkin method for solving the Burgers equation", Int. J. Comput. Math., 81(11), pp. 1433-1444 (2004).

25. Dag, I., Saka, B., and Boz, A. "B-spline Galerkin methods for numerical solutions of Burgers' equation", Appl. Math. Comput., 166, pp. 506-522 (2005).

26. Aksan, E.N. "An application of cubic B-spline finite element method for the Burgers' equation", Therm. Sci., 22(1), pp. 195-202 (2018).

27. Ak, T. "An application of Galerkin method to generalized Benjamin-Bona-Mahony-Burgers equation", Adiyaman University Journal of Science, 8(2), pp. 53- 69 (2018).

28. Ak, T., Saha, A., and Dhawan, S. "Performance of a hybrid computational scheme on travelling waves and its dynamic transition for Gilson Pickering equation", Int. J. Mod. Phys. C, 30(4), pp. 1950028-1/17 (2019).

29. Mohammadi, R. "Numerical approximation for viscous Cahn-Hilliard equation via septic B-spline", Appl. Anal. (2019). DOI: 10.1080/00036811.2019.1594200.

30. Mukundan, V. and Awasthi, A. "Efficient numerical techniques for Burgers' equation", Appl. Math. Comput., 262, pp. 282-297 (2015).

31. Aksan, E.N. "Quadratic B-spline finite element method for numerical solution of the Burgers equation", Appl. Math. Comput., 174, pp. 884-896 (2006).

32. Tunc, H. "Various finite element techniques for advection-diffusion-reaction processes", MSc Thesis, Yildiz Technical University (2017).

33. Shao, L., Feng, X., and He, Y. "The local discontinuous Galerkin finite element method for Burgers equation", Math. Comput. Model., 54, pp. 2943-2954 (2011).

34. Abazari, R. and Borhanifar, A. "Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method", Comput. Math. Appl., 59, pp. 2711-2722 (2010).

2. Wang, J. and Warnecke, G. "Existence and uniqueness of solutions for a non-uniformly parabolic equation", J. Differ. Equations, 189, pp. 1-16 (2003).

3. Miller, E.L. "Predictor-corrector studies of Burger's model of turbulent flow", MS Thesis, University of Delaware, Newark, Delaware (1966).

4. Kutluay, S., Bahadir, A.R., and Ozdes, A. "Numerical solution of one-dimensional Burgers' equation: explicit and exact-explicit finite difference methods", J. Comput. Appl. Math., 103, pp. 251-261 (1999).

5. Seydaoglu, M. "An accurate approximation algorithm for Burgers' equation in the presence of small viscosity", J. Comput. Appl. Math., 344, pp. 473-481 (2018).

6. Bahadir, A.R. and Saglam, M. "A mixed finite difference and boundary element approach to onedimensional Burgers' equation", Appl. Math. Comput., 160, pp. 663-673 (2005).

7. Sari, M., Tunc, H., and Seydaoglu M. "Higher order splitting approaches in analysis of the Burgers equation", Kuwait J. Sci., 46(1), pp. 1-14 (2019).

8. Hopf, E. "The partial differential equation", Commun. Pur. Appl. Math., 9, pp. 201-230 (1950).

9. Cole, J.D. "On a quasi-linear parabolic equation in aerodynamics", Q. Appl. Math., 9, pp. 225-236 (1951).

10. Talwar, J., Mohanty, R.K., and Singh, S. "A new algorithm based on spline in tension approximation for 1D parabolic quasi-linear equations on a variable mesh", Int. J. Comput. Math., 93, pp. 1771-1786 (2016).

11. Jiwari, R. "A hybrid numerical scheme for the numerical solution of the Burgers' equation", Comput. Phys. Commun., 188, pp. 50-67 (2015).

12. Seydaoglu, M., Erdogan, U., and Ozis, T. "Numerical solution of Burgers' equation with higher order splitting methods", J. Comput. Appl. Math., 291, pp. 410-421 (2016).

13. Korkmaz, A. and Dag, I. "Polynomial based differential quadrature method for numerical solution of nonlinear Burgers' equation", J. Franklin I., 248, pp. 2863-2875 (2016).

14. Bahadir, A.R. and Saglam, M. "A mixed finite difference and boundary element approach to onedimensional Burgers' equation", Appl. Math. Comput., 160, pp. 663-673 (2005).

15. Egidi, N., Maponi, P., and Quadrini, M. "An integral equation method for the numerical solution of the Burgers equation", Comput. Math. Appl., 76(1), pp. 35-44 (2018).

16. Inan, B. and Bahadir, A.R. "An explicit exponential finite difference method for the Burgers' equation", European International Journal of Science and Technology, 2(10), pp. 61-72 (2013).

17. Zeytinoglu, A., Sari, M., and Pasaoglu, B.A. "Numerical simulations of shock wave propagating by a hybrid approximation based on high-order finite difference schemes", Acta Phys. Pol. A, 133, pp. 369-386 (2017).

18. Verma, A.K. and Verma, L. "Higher order time integration formula with application on Burgers' equation", Int. J. Comput. Math., 92, pp. 756-771 (2015).

19. Rouzegar, J. and Sharifpoor, R.A. "A finite element formulation for bending analysis of isotropic and orthotropic plates based on two-variable refined plate theory", Sci. Iran., 22(1), pp. 196-207 (2015).

20. Ak, T., Karakoc, S.B.G., and Biswas, A. "Application of Petrov-Galerkin finite element method to shallow water waves model: modified Korteweg-de Vries equation", Sci. Iran., 24(3), pp. 1148-1159 (2017).

21. Zendehbudi, G. "Combination of projection and Galerkin finite element methods to solve the problem of free convection in enclosures with complex geometries", Sci. Iran, 25(3), pp. 1189-1196 (2018).

22. Kutluay, S., Esen, A., and Dag, I. "Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method", J. Comput. Appl. Math., 167, pp. 21-33 (2004).

23. Sari, M. and Tunc, H. "An optimization technique in analyzing the Burgers equation", Sigma Journal of Engineering and Natural Sciences, 35(3), pp. 369-386 (2017).

24. Kutluay, S. and Esen, A. "A lumped Galerkin method for solving the Burgers equation", Int. J. Comput. Math., 81(11), pp. 1433-1444 (2004).

25. Dag, I., Saka, B., and Boz, A. "B-spline Galerkin methods for numerical solutions of Burgers' equation", Appl. Math. Comput., 166, pp. 506-522 (2005).

26. Aksan, E.N. "An application of cubic B-spline finite element method for the Burgers' equation", Therm. Sci., 22(1), pp. 195-202 (2018).

27. Ak, T. "An application of Galerkin method to generalized Benjamin-Bona-Mahony-Burgers equation", Adiyaman University Journal of Science, 8(2), pp. 53- 69 (2018).

28. Ak, T., Saha, A., and Dhawan, S. "Performance of a hybrid computational scheme on travelling waves and its dynamic transition for Gilson Pickering equation", Int. J. Mod. Phys. C, 30(4), pp. 1950028-1/17 (2019).

29. Mohammadi, R. "Numerical approximation for viscous Cahn-Hilliard equation via septic B-spline", Appl. Anal. (2019). DOI: 10.1080/00036811.2019.1594200.

30. Mukundan, V. and Awasthi, A. "Efficient numerical techniques for Burgers' equation", Appl. Math. Comput., 262, pp. 282-297 (2015).

31. Aksan, E.N. "Quadratic B-spline finite element method for numerical solution of the Burgers equation", Appl. Math. Comput., 174, pp. 884-896 (2006).

32. Tunc, H. "Various finite element techniques for advection-diffusion-reaction processes", MSc Thesis, Yildiz Technical University (2017).

33. Shao, L., Feng, X., and He, Y. "The local discontinuous Galerkin finite element method for Burgers equation", Math. Comput. Model., 54, pp. 2943-2954 (2011).

34. Abazari, R. and Borhanifar, A. "Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method", Comput. Math. Appl., 59, pp. 2711-2722 (2010).

Transactions on Mechanical Engineering (B)

November and December 2020Pages 2853-2870