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

Pedro Henrique de Almeida Konzen, F S Azevedo, E Sauter, P R A Zingano

Abstract


In this work we present an efficient Galerkin least squares finite element scheme to simulate the Burgers' equation on the whole real line and subjected to initial conditions with compact support. The numerical simulations are performed by considering a sequence of auxiliary spatially dimensionless Dirichlet's problems parameterized by its numerical support $\tilde{K}$. Gaining advantage from the well-known convective-diffusive effects of the Burgers' equation, computations start by choosing $\tilde{K}$ so it contains the support of the initial condition and, as solution diffuses out, $\tilde{K}$ is increased appropriately. By direct comparisons between numerical and analytic solutions and its asymptotic behavior, we conclude that the proposed scheme is accurate even for large times, and it can be applied to numerically investigate properties of this and similar equations on unbounded domains.

Keywords


Burgers' equation on the real line; Galerkin least squares finite element method; asymptotic properties

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 Loading ...

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:

                       

         

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia