### Numerical Simulations with the Galerkin Least Squares Finite Element Method for the Burgers' Equation on the Real Line

#### Abstract

#### Keywords

#### Full Text:

PDF#### References

E.N. Aksan. A numerical solution of Burgers' equation by finite element method constructed on the method of discretization in time. Appl. Math. Comput., 170:895--904, 2005.

E.N. Aksan. Quadratic B-spline finite element method for numerical solution of the Burgers' equation.. Appl. Math. Comput., 174:884--896, 2006.

E.N. Aksan and A.Özdecs. A numerical solution of Burgers' equation. Appl. Math. Comput., 156:395--402, 2004.

P. Arminjon and C. Beauchamp. Continuous and discontinuous finite element methods for Burgers' equation. Comput. Methods Appl. Mech. Engrg., 25:65--84, 1981.

W. Bangerth, D. Davydov, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler,

M. Maier, B. Turcksin, and D. Wells. The deal.II library, version 8.47. Journal of Numerical Mathematics, 24, 2016.

W. Bangerth, R. Hartmann, and G. Kanschat. deal.II -- a general purpose object oriented finite element library. ACM Trans. Math. Softw., 33(4):24/1--24/27, 2007.

C. Basdevant, M. Deville, P. Haldenwang, J.M. Lacroix, J. Quazzani, R. Peyret, and P.~Orlandi. Spectral and finite difference solutions of the Burgers' equation. Comput Fluids, 14:23--41, 1986.

M. Basto, V. Semiao, and F. Calheiros. Dynamics in spectral solutions of Burgers equation. J. Comput. Appl. Math., 205:296--304, 2006.

M.A.H. Bateman. Some recent researches on the motion of fluids. Mon. Wea. Rev., 43:163--170, 1915.

Malte Braack. Finite elemente. available online, http://www.numerik.uni-kiel.de/~mabr/lehre/skripte/fem-braack.pdf, Jan. 2015.

J.M. Burgers. The nonlinear diffusion equation. Springer, 1974.

Fletcher C.A., Numerical Solutions of Partial Differential Equations, chapter

Burgers’ equation: a model for all reasons, pages 139--225. Nort-Holland, Amsterdam, 1982.

J. Caldwell, R. Saunders, and P. Wanless. A note on variation-iterative schemes applied to Burgers' equation. J. Comput. Phys., 58:275--281, 1985.

J. Caldwell and P. Smith. Solution of Burgers' equation with a large Reynolds number. Appl. Math. Modelling, 6:381--385, 1982.

J. Caldwell, P. Wanless, and A.E. Cook. A finite element approach to Burgers' equation. Appl. Math. Modelling, 5:189--193, 1981.

J. Caldwell, P. Wanless, and A.E. Cook. Solution of Burgers' equation for large Reynolds number using finite elements with moving nodes. Appl. Math. Modelling, 11:211--214, 1987.

Julian D Cole et al. On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math, 9(3):225--236, 1951.

A. Dogan. A Galerkin finite element method to Burgers' equation. Appl. Math. Comput., 157:331--346, 2004.

M.B. Abd el Malek and S.M.A. El-Mansi. Group theoretic methods applied to Burgers' equation. J. Comput. Appl. Math., 115:1--12, 2000.

L.C. Evans. Partial differential equations, volume 19 of Graduate

Studies in Mathematics. The American Mathematical Society, 2nd edition, 2010.

C.A.J. Fletcher. A comparison of finite element and finite difference solutions of the one- and two-dimensional Burgers' equations. J. Comput. Phys., 51:159--188, 1983.

A.~Gorguis. A comparision between Cole-Hopf transformation and the decompisition method for solving Burgers' equations. Appl. Math. Comput., 173:126--136, 2006.

M. Gülsu. A finite difference approach for solution of Burgers' equation.

Appl. Math. Comput., 175:1245--1255, 2006.

A. Hashemian and H. Shodja. A meshless approach for solution of Burgers' equation. J. Comput. Appl. Math., 220:226--239, 2008.

C.J. Holland. On the limiting behavior of Burger's equation. J. Math. Anal. Appl., 57:156--160, 1977.

Y.C. Hon and X.Z. Mao. An efficient numerical scheme for Burgers' equation.

J. Comput. Appl. Math., 95:37--50, 1998.

E.~Hopf. The partial differential equation u_t + uu_x = mu u_{xx}. Comm. Pure and Appl. Math., 3:201--230, 1950.

A.N. Hrymak, G.J. McRae, and A.W. Westerberg. An implementation of a moving finite element method. J. Comput. Phys., 63:168--190, 1986.

M. Inc. On numerical solution of Burgers' equation by homotopy analysis

method. J. Phys. A, 372:356--360, 2008.

R.~Jiwari. A hybrid numerical scheme for the numerical solution of the Burgers' equation. Comput. Phys. Commun., 188:59--67, 2015.

C. Johnson. Numerical solutions of partial differential equations by the

finite element method. Dover, 2009.

M. Kadalbajoo and A. Awasthi. A numerical method based on Crank-Nicolson scheme for Burgers' equation. Appl. Math. Comput., 182:1430--1442, 2006.

C.T. Kelley. Solving nonlinear equations with the Newton's method.

SIAM, 2003.

A.H. Khater, R.S. Temsah, and M.M. Hassan. A Chebyshev spectral collocation method for solving Burgers'-type equations. J. Comput. Appl. Math., 222:333--350, 2008.

S. Kutluay, A.R. Bahadir, and A. Özdecs.

Numerical solution of one-dimensional Burgers equation: explicit and

exact-explicit finite difference methods. J. Comput. Appl. Math., 103:251--261, 1999.

S. Kutluay, A. Esen, and I. Dag. Numerical solutions of the Burgers' equation by the least-squares quadratic {B}-spline finite element method.

J. Comput. Appl. Math., 167:21--33, 2004.

C.A. Ladeia, N.M. Romero, P.L. Natti, and E.R. Cirilo. Formulações semi-discretas para a equação 1d de Burgers. TEMA (São Carlos), 14(3):319 -- 331, 2013.

Mats G. Larson and Fredrik Bengzon. The Finite Element Method: Theory, Implementation, and Applications. Springer-Verlag Berlin Heidelberg, 2013.

V. Mukundan and A. Awasthi. Efficient numerical techniques for Burgers' equation. Appl. Math. Comput., 262:282--297, 2015.

T. Özics, E.N. Aksan, and A.~Özdecs. A finite element approach for solution of Burgers' equation. Appl. Math. Comput., 139:417--428, 2003.

T. Özics and Y. Aslan. The semi-approximate approach for solving Burgers' equation with high Reynolds number. Appl. Math. Comput., 163:131--145, 2005.

T. Özics, A. Esen, and S. Kutluay. Numerical solution of Burgers' equation by quadratic {B}-spline finite elements. Appl. Math. Comput., 165:237--249, 2005.

T. Ozis and A. Ozdes. A direct variational methods applied to Burgers' equation. J. Comput. Appl. Math., 71:163--175, 1996.

E.Y. Rodin. On some approximate and exact solutions of boundary value problems for Burgers' equation. J. Math. Anal. Appl., 30:401--414, 1970.

B. Saka and I. Daug. A numerical study of the Burgers' equation.

J. Frankl. Inst., 345:328--348, 2008.

L. Shao, X. Feng, and Y. He. The local discontinuous Galerkin finite element method for Burgers' equation. Math. Comput. Model., 54:2943--2954, 2011.

A.H.A.E. Tabatabaei, E. Shakour, and M. Dehghan. Some implicit methods for the numerical solution of Burgers' equation. Appl. Math. Comput., 191:560--570, 2007.

W.L. Wood. An exact solution for Burger's equation. Commun. Numer. Meth. Engng., 22:797--798, 2006.

M. Xu, R.-H. Wang, J.-H. Zhang, and Q.~Fang. A novel numerical scheme for solving Burgers' equation. Appl. Math. Comput., 217:4473--4482, 2011.

X.H. Zhang, J. Ouyang, and L. Zhang. Element-free characteristic Galerkin method for Burgers' equation. Eng. Anal. Boundary Elem., 33:356--362, 2009.

P.R. Zingano. Some asymptotic limits for solutions of Burgers equation.

available at: url{http://arxiv.org/pdf/math/0512503.pdf}, 1997. Universidade Federal do Rio Grande do Sul.

DOI: https://doi.org/10.5540/tema.2017.018.02.0287

#### Article Metrics

_{Metrics powered by PLOS ALM}

### Refbacks

- There are currently no refbacks.

**Trends in Computational and Applied Mathematics**

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

Indexed in: