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April  2013, 33(4): 1275-1291. doi: 10.3934/dcds.2013.33.1275

On a variational approach for the analysis and numerical simulation of ODEs

Sergio Amat 1, and  Pablo Pedregal 2,

1. 

Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Spain
2. 

E.T.S. Ingenieros Industriales, Universidad de Castilla La Mancha

Received  October 2011 Revised  March 2012 Published  October 2012

This paper is devoted to the study and approximation of systems of ordinary differential equations based on an analysis of a certain error functional associated, in a natural way, with the original problem. We prove that in seeking to minimize the error by using standard descent schemes, the procedure can never get stuck in local minima, but will always and steadily decrease the error until getting to the original solution. One main step in the procedure relies on a very particular linearization of the problem: in some sense, it is like a globally convergent Newton type method. Although our objective here is not to perform a rigorous numerical study of the method, we illustrate its potential for approximation by considering some stiff systems of equations. The performance is astonishingly very good due to the fact that we can use very robust methods to approximate linear stiff problems like implicit collocation schemes. We also include a couple of typical test models for the Lorentz system and the Kepler problem, again confirming a very good performance. We believe that this approach can be used in a systematic way to examine other situations and other types of equations due to its flexibility and its simplicity.
Keywords: variational methods, optimality conditions, ODEs, approximation..
Mathematics Subject Classification: Primary: 34A34, 49M05; Secondary: 65L04, 65L0.
Citation: Sergio Amat, Pablo Pedregal. On a variational approach for the analysis and numerical simulation of ODEs. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1275-1291. doi: 10.3934/dcds.2013.33.1275
References:
[1]

S. Amat and P. Pedregal, A variational approach to implicit ODEs and differential inclusions, ESAIM-COCV, 15 (2009), 139-148.  Google Scholar

[2]

S. Amat, D. J. López and P. Pedregal, Numerical approximation to ODEs using a variational approach I: The basic framework,, to appear in Optimization., ().   Google Scholar

[3]

W. Auzinger, R. Frank and G. Kirlinger, An extension of $B$-convergence for Runge-Kutta methods, Appl. Num. Math., 9, (1992), 91-109.  Google Scholar

[4]

W. Auzinger, R. Frank and G. Kirlinger, Modern convergence theory for stiff initial value problems, J. Comput. Appl. Math., 45 (1993), 5-16. doi: 10.1016/0378-3782(93)90046-W.  Google Scholar

[5]

G. D. Byrne and A. C. Hindmarsh, Stift ODE solvers: A review of current and coming attractions, J. Comput. Phys., 70 (1987), 1-62.  Google Scholar

[6]

G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532.  Google Scholar

[7]

J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin, Germany, 1980.  Google Scholar

[8]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations," Springer Series in Computational Mathematics, Springer, 2006.  Google Scholar

[9]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems," Springer-Verlag, Berlin, Germany, 1991.  Google Scholar

[10]

J. D. Lambert, "Numerical Methods for Ordinary Differntial Systems: The Initial Value Problem," John Wiley and Sons Ltd. 1991.  Google Scholar

[11]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators, Internat. J. Numer. Methods Engrg., 60 (2004), 153-212.  Google Scholar

[12]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  Google Scholar

[13]

The MathWorks, Inc., MATLAB and SIMULINK,, Natick, ().   Google Scholar

[14]

P. Pedregal, A variational approach to dynamical systems, and its numerical simulation, Numer. Funct. Anal. Opt., 31 (2010), 1532-2467. doi: 10.1080/01630563.2010.497237.  Google Scholar

[15]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," Springer-Verlag. Second edition, 1993.  Google Scholar

show all references

References:
[1]

S. Amat and P. Pedregal, A variational approach to implicit ODEs and differential inclusions, ESAIM-COCV, 15 (2009), 139-148.  Google Scholar

[2]

S. Amat, D. J. López and P. Pedregal, Numerical approximation to ODEs using a variational approach I: The basic framework,, to appear in Optimization., ().   Google Scholar

[3]

W. Auzinger, R. Frank and G. Kirlinger, An extension of $B$-convergence for Runge-Kutta methods, Appl. Num. Math., 9, (1992), 91-109.  Google Scholar

[4]

W. Auzinger, R. Frank and G. Kirlinger, Modern convergence theory for stiff initial value problems, J. Comput. Appl. Math., 45 (1993), 5-16. doi: 10.1016/0378-3782(93)90046-W.  Google Scholar

[5]

G. D. Byrne and A. C. Hindmarsh, Stift ODE solvers: A review of current and coming attractions, J. Comput. Phys., 70 (1987), 1-62.  Google Scholar

[6]

G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532.  Google Scholar

[7]

J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin, Germany, 1980.  Google Scholar

[8]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations," Springer Series in Computational Mathematics, Springer, 2006.  Google Scholar

[9]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems," Springer-Verlag, Berlin, Germany, 1991.  Google Scholar

[10]

J. D. Lambert, "Numerical Methods for Ordinary Differntial Systems: The Initial Value Problem," John Wiley and Sons Ltd. 1991.  Google Scholar

[11]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators, Internat. J. Numer. Methods Engrg., 60 (2004), 153-212.  Google Scholar

[12]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  Google Scholar

[13]

The MathWorks, Inc., MATLAB and SIMULINK,, Natick, ().   Google Scholar

[14]

P. Pedregal, A variational approach to dynamical systems, and its numerical simulation, Numer. Funct. Anal. Opt., 31 (2010), 1532-2467. doi: 10.1080/01630563.2010.497237.  Google Scholar

[15]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," Springer-Verlag. Second edition, 1993.  Google Scholar

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