American Institute of Mathematical Sciences

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

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.
Citation: Sergio Amat, Pablo Pedregal. On a variational approach for the analysis and numerical simulation of ODEs. Discrete and 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. [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., (). [3] W. Auzinger, R. Frank and G. Kirlinger, An extension of $B$-convergence for Runge-Kutta methods, Appl. Num. Math., 9, (1992), 91-109. [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. [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. [6] G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532. [7] J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin, Germany, 1980. [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. [9] E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems," Springer-Verlag, Berlin, Germany, 1991. [10] J. D. Lambert, "Numerical Methods for Ordinary Differntial Systems: The Initial Value Problem," John Wiley and Sons Ltd. 1991. [11] A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators, Internat. J. Numer. Methods Engrg., 60 (2004), 153-212. [12] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. [13] The MathWorks, Inc., MATLAB and SIMULINK,, Natick, (). [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. [15] J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," Springer-Verlag. Second edition, 1993.

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References:
 [1] S. Amat and P. Pedregal, A variational approach to implicit ODEs and differential inclusions, ESAIM-COCV, 15 (2009), 139-148. [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., (). [3] W. Auzinger, R. Frank and G. Kirlinger, An extension of $B$-convergence for Runge-Kutta methods, Appl. Num. Math., 9, (1992), 91-109. [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. [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. [6] G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532. [7] J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin, Germany, 1980. [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. [9] E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems," Springer-Verlag, Berlin, Germany, 1991. [10] J. D. Lambert, "Numerical Methods for Ordinary Differntial Systems: The Initial Value Problem," John Wiley and Sons Ltd. 1991. [11] A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators, Internat. J. Numer. Methods Engrg., 60 (2004), 153-212. [12] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. [13] The MathWorks, Inc., MATLAB and SIMULINK,, Natick, (). [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. [15] J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," Springer-Verlag. Second edition, 1993.
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