Advertisement

On the presumed superiority of analytical solutions over numerical methods

  • Vincent ArdourelEmail author
  • Julie Jebeile
Original Paper in Philosophy of Science
First Online:

Abstract

An important task in mathematical sciences is to make quantitative predictions, which is often done via the solution of differential equations. In this paper, we investigate why, to perform this task, scientists sometimes choose to use numerical methods instead of analytical solutions. Via several examples, we argue that the choice for numerical methods can be explained by the fact that, while making quantitative predictions seems at first glance to be facilitated by analytical solutions, this is actually often much easier with numerical methods. Thus we challenge the widely presumed superiority of analytical solutions over numerical methods.

Keywords

Applied mathematics Exactness Analytical solutions Numerical methods 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

We thank the anonymous reviewers for their helpful comments, and their contribution to enhancing the quality of this paper. This work was supported by French State funds managed by the National Research Agency on the behalf of Idex Sorbonne Universités within the Investissements d’Avenir Programme under reference ANR-11-IDEX-0004-02. One of the authors is currently a beneficiary of a “MOVE-IN Louvain” Incoming Post-doctoral Fellowship, co-funded by the Marie Curie Actions of the European Commission.

References

  1. Babelon, O., Bernard, D., & Talon, M. (2003). Introduction to classical integrable systems. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  2. Barberousse, A., & Imbert, C. (2014). Recurring models and sensitivity to computational constraints. The Monist, 97(3), 259–279.CrossRefGoogle Scholar
  3. Barberousse, A., Franceschelli, S., & Imbert, C. (2009). Computer simulations as experiments. Synthese, 169(3), 557–574.CrossRefGoogle Scholar
  4. Batterman, R.W. (2007). On the specialness of special functions (The nonrandom effusions of the divine mathematician). The British Journal for the Philosophy of Science, 58(2), 263–286.CrossRefGoogle Scholar
  5. Belendez, A., Pascual, C., Mendez, D.I., Belendez, T., & Neipp, C. (2007). Exact solution for the nonlinear pendulum. Revista Brasileira de Ensino de Fisica, 29(4), 645–648.CrossRefGoogle Scholar
  6. Borwein, J., & Crandall, R. (2013). Closed forms: what they are and why we care. Notices of the American Mathematical Society, 60(1), 50–65.CrossRefGoogle Scholar
  7. Corless, R.M., & Fillion, N. (2014). A graduate introduction to numerical methods. Springer.Google Scholar
  8. Diacu, F. (1996). The solution of the N-boby problem. The Mathematical Intelligencer, 18(3), 66–70.CrossRefGoogle Scholar
  9. Dutt, R. (1976). Application of Hamilton-Jacobi theory to the Lotka-Volterra oscillator. Bulletin of Mathematical Biology, 38, 459–465.CrossRefGoogle Scholar
  10. Einstein, T.L. (2003). Applications of ideas from random matrix theory to step distributions on “misoriented” surfaces. Annales Henri Poincaré, 4(Suppl. 2), 811–824.CrossRefGoogle Scholar
  11. Evans, C.M., & Findley, G.L. (1999). Analytic solutions to a family of Lotka–Volterra related differential equations. Journal of Mathematical Chemistry, 25, 181–189.CrossRefGoogle Scholar
  12. Fillion, N., & Corless, R.M. (2014). On the epistemological analysis of modeling and computational error in the mathematical sciences. Synthese, 191, 1451–1467.CrossRefGoogle Scholar
  13. Fillion, N., & Bangu, S. (2015). Numerical methods, complexity and epistemic hierarchies. Philosophy of Science. forthcoming.Google Scholar
  14. Forsythe, G.E. (1970). Pitfalls in computation, or why a math book isn’t enough. The American Mathematical Monthly, 77(9), 931–956.CrossRefGoogle Scholar
  15. French, A. P., & Taylor, E.F. (1998). An introduction to quantum physics. Cheltenham: Stanley Thomas.Google Scholar
  16. Gallant, J. (2012). Doing physics with scientific notebook. Wiley.Google Scholar
  17. Goldstein, H., Poole, C., & Safko, J. (2001). Classical mechanics.Google Scholar
  18. Goriely, A. (2001). Integrability and nonintegrability of dynamical systems. Advanced series in nonlinear dynamics Vol. 19. World Scientific.Google Scholar
  19. Hartmann, S. (1996). The world as a process - Simulations in the natural and social sciences. In R.U.M. Hegselmann, & K. Troitzsch (Eds.), Modelling and simulation in the social sciences from the philosophy of science point of view (pp. 77–100). Dordrecht: Kluwer.Google Scholar
  20. Hairer, E., Nørsett, S.P., & Wanner, G. (1992). Solving ordinary differential equations I: Nonstiff problems. Springer.Google Scholar
  21. Hiestand, J.W. (2009). Numerical methods with VBA programming. John & Bartlett Publishers.Google Scholar
  22. Henkel, M. (2001). Sur la solution de Sundman du problme des trois corps. Philosophia Scientiae, 5(2), 161–184.Google Scholar
  23. Humphreys, P. (2004). Extending ourselves: Computational science, empiricism, and scientific method. New-York: Oxford University Press.CrossRefGoogle Scholar
  24. Humphreys, P. (2009). The philosophical novelty of computer simulation methods. Synthese, 169(3), 615–626.CrossRefGoogle Scholar
  25. Masoliver, J., & Ros, A. (2011). Integrability and chaos: The classical uncertainty. European Journal of Physics, 32, 431–458.CrossRefGoogle Scholar
  26. Murray, J. (2002). Mathematical biology. An Introduction Vol. 1. Springer.Google Scholar
  27. Morrison, M. (2009). Models, measurement and computer simulation: The changing face of experimentation. Philosophical Studies, 143, 33–57.CrossRefGoogle Scholar
  28. Ortega, J.M. (1992). Numerical analysis. Ed. SIAM.Google Scholar
  29. Singer, M.F. (1990). Formal solutions of differential equations. Journal of Symbolic Computation, 10, 59–94.CrossRefGoogle Scholar
  30. Stern, A., & Desbrun, M. (2008). Discrete geometric mechanics for variational time integrators. Siggraph 2006 Course Notes, chap. 15.Google Scholar
  31. Stoer, J., & Bulisch, R. (2002). Introduction to numerical analysis, Texts in Applied Mathematics Vol. 12. Springer.Google Scholar
  32. Süli, E., & Mayers, D.F. (2003). An introduction to numerical analysis. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  33. Sundman, K. (1907). Recherches sur le problème des trios corps. Acta Societatis Scientiarum Fennicae, 34(6).Google Scholar
  34. Sundman, K. (1909). Nouvelles recherches sur le problème des trois corps. Acta Societatis Scientiarum Fennicae, 35(9).Google Scholar
  35. Wang, Q.-D. (1991). The global solution of the n-body problem. Celestial Mechanics and Dynamics Astronomy, 50, 73–88.CrossRefGoogle Scholar
  36. Zoladek, H. (1998). The extented monodromy group and liouvillian first intergrals. Journal of Dynamical and Control Systems, 4(1), 1–28.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institut Supérieur de PhilosophieUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Sciences, Normes, Décision (FRE 3593 CNRS, Université Paris-Sorbonne)ParisFrance

Personalised recommendations

Cite article

Buy options