Formulações Semi-Discretas para a Equação 1D de Burgers
DOI:
https://doi.org/10.5540/tema.2013.014.03.0319Abstract
Neste trabalho fizemos comparações entre formulações semi-discretas para a obtenção de soluções numéricas para a equação 1D de Burgers. As formulações consistem em discretizar o domínio temporal via métodos implícitos multiestágios de segunda e quarta ordem: aproximantes de Padé R11 e R22; e o domínio espacial via métodos de elementos finitos: mínimos quadrados (MEFMQ), Galerkin (MEFG) e Streamline-Upwind Petrov-Galerkin (SUPG). Conhecendo as soluções analíticas da equação 1D de Burgues, para diferentes condições iniciais e de fronteira, foram realizadas análises dos erros numéricos a partir das normas L2 e Linf. Verificamos que o método com o aproximante de Padé R22 adicionado as formulações MEFMQ, MEFG e SUPG, aumentou a região de convergência das soluções numéricas e apresentou maior precisão quando comparado as soluções obtidas por meio do aproximante de Padé R11. Constamos que o método R22 amenizou as oscilações das soluções numéricas associadas as formulações MEFG e SUPG.
References
D. Behmardi, D.E. Nayeri, Introduction of Fréchet and Gâteaux Derivative. Appl. Math. Sci., 2 (2008) 975-980.
C.S. Brenner, R.L. Scott, “The Mathematical Theory of Finite Element Methods”, New York, Springer-Verlag, 2008.
A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulationsfor convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation,Comput. Meth. Appl. Mech. Eng., 32 (1982) 199-259.
G.P. Ciarlet, “The Finite Element Method for Elliptic Problems”. North Holland, SIAM, 1978.
E.A. David, B. Oscar, Time stepping via one-dimensional Padé approximation, J. Sci. Comput., 30 (2005), 83-115.
A.A. Dogan, T.J.R. Hughes, A Galerkin element approach to Burgers’ equations, Applied Mathematics and Computation, 154 (2004) 331-346.
J. Donea, B. Roig, A. Huerta, “Finite Element Methods for Flow Problems”.John Wiley and Sons, Chichester, 2003.
J. Donea, B. Roig, A. Huerta, Higher-order accurate time-stepping schemes for convection-diffusion problems, Comput. Meth. Appl. Mech. Engng., 182 (2000) 249-275.
V.G. Ferreira, G.A.B. Lima, L. Corrêa, A.C. Cansezano, E.R. Cirilo, P.L. Natti,N.M.L. Romeiro, Avaliação computacional, de esquemas convectivos em problemas
de dinâmica dos fluidos. Semina: Ciências Tecnológicas, 2 (2012) 107-116.
V.G. Ferreira, R.A.B. de Queiroz, G.A.B. Lima, R.G. Cuenca, C.M. Oishi, J.L.F. Azevedo, S. McKee, A bounded upwinding scheme for computing convection-dominated transport problems, 57 (2012) 208-224.
E. Hairer, S.P. Norsett, G. Wanner, “Solving ordinary differential equations I, Non-stiff Problems”, New York, Springer-Verlag, 1987.
A. Huerta, B. Roig, J. Donea, Time-accurate solution of stabilized convectiondiffusion-reaction equations: II - accuracy analysis and examples, Commun. Numer. Meth. Eng., 18 (2002) 575-584
P.C. Jain, R. Shankar, T.V. Singh, Numerical Technique for Solving Convection-Reaction-Diffusion Equation, Math. Comput. Model, 22 (1995) 113-
B.N, Jiang, “The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics”. Berlin: Springer,
K. Kakuda, N. Tosaka, The generalised boundary element approach to Burgers equation, Int. J. Numer. Meth. Eng., 29 (1990) 245-261.
S. Kutluay, A. Esen, I. Dag, Numerical solutions of the Burgers equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math., 167 (2004) 21-33.
C.A. Ladeia, “Formulação semi-discreta aplicada as equações 1D de convecçãodifusão-reação e de Burger”, Dissertação de Mestrado, PGMAC/UEL, Londrina,
Pr, 2012.
J.D. Lambert, “Numerical Methods for Ordinary Differential Systems”, New York, Wiley, 1993.
J.T. Oden, T. Belytschko, I. Babuska, T.J.R. Hughes, Research directions in computational mechanics, Comput. Methods Appl. Mech. Eng., 192 (2003) 913-922.
S.R. Pardo, P.L. Natti, N.M.L. Romeiro, E.R. Cirilo, A transport modeling of the carbon-nitrogen cycle at Igapó I Lake - Londrina, Paraná State, Brazil, Acta
Scientiarum, Technology, 2 (2012) 217-226.
N.M.L. Romeiro, R.S.G Castro, S.M.C. Malta, L. Landau, A linearization technique for multi-species transport problems, Trans. Porous Med., 70 (2007)
-10.
P.L. Sachdev, “Nonlinear Diffusive Waves”, Cambridge University Press, Cambridge, 1987.
J.C. Strikwerda, “Finite Difference Schemes and Partial Differential Equations”, SIAM, 2004.
Z.F. Tian, P.X. Yu, A High-order exponencial scheme for solving 1D unsteady convection-difusion equations, Jornal of Computational and Applied Mathematics,
(2011), 2477-2491.
Downloads
Additional Files
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.
