, 56:12 | Cite as

Optimality and resonances in a class of compact finite difference schemes of high order

  • Joackim BernierEmail author


In this paper, we revisit the old problem of compact finite difference approximations of the homogeneous Dirichlet problem in dimension 1. We design a large and natural set of schemes of arbitrary high order, and we equip this set with an algebraic structure. We give some general criteria of convergence and we apply them to obtain two new results. On the one hand, we use Padé approximant theory to construct, for each given order of consistency, the most efficient schemes and we prove their convergence. On the other hand, we use diophantine approximation theory to prove that almost all of these schemes are convergent at the same rate as the consistency order, up to some logarithmic correction.


Finite difference High order Padé approximant Dirichlet problem Resonances 

Mathematics Subject Classification




  1. 1.
    Baker Jr., G.A., Graves-Morris, P.: Padé Approximants, Encyclopedia of Mathematics and its Applications, vol. 59, 2nd edn. Cambridge University Press, Cambridge (1996). CrossRefGoogle Scholar
  2. 2.
    Borwein, J.M., Chamberland, M.: Integer powers of arcsin. Int. J. Math. Math. Sci. (2007).
  3. 3.
    Bramble, J.H., Hubbard, B.E.: New monotone type approximations for elliptic problems. Math. Comput. 18, 349–367 (1964). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Coulombel, J.F.: On the strong stability of finite difference schemes for hyperbolic systems in two space dimensions. Calcolo 51(1), 97–108 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fornberg, B.: Generation of finite difference formulas on arbitrarily spaced grids. Math. Comput. 51(184), 699–706 (1988). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gilewicz, J.: Approximants de Padé. Lecture Notes in Mathematics, vol. 667. Springer, Berlin (1978)Google Scholar
  7. 7.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Springer Series in Computational Mathematics, vol. 31. Springer, Heidelberg (2010). Structure-preserving algorithms for ordinary differential equations, Reprint of the second edition (2006)Google Scholar
  8. 8.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, 2nd edn. Springer, Berlin (1993)Google Scholar
  9. 9.
    Kanazawa, H., Matsuo, T., Yaguchi, T.: A conservative compact finite difference scheme for the KdV equation. JSIAM Lett. 4, 5–8 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karp, D., Prilepkina, E.: Hypergeometric functions as generalized Stieltjes transforms. J. Math. Anal. Appl. 393(2), 348–359 (2012).
  11. 11.
    Merrien, J.L.: Approximation de problèmes faiblement non linéaires par des schémas de différences finies superconvergents. Ph.D. thesis, Université de Rennes \(1\) (1985)Google Scholar
  12. 12.
    Noumerov, B.V.: A method of extrapolation of perturbations. Mon. Not. R. Astron. Soc. 84, 592 (1924)CrossRefGoogle Scholar
  13. 13.
    Price, H.S.: Monotone and oscillation matrices applied to finite difference approximations. Math. Comput. 22, 489–516 (1968). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shidlovskii, A.B.: Transcendental numbers, De Gruyter Studies in Mathematics, vol. 12. Walter de Gruyter & Co., Berlin (1989). Translated from the Russian by Neal Koblitz, With a foreword by W. Dale Brownawell
  15. 15.
    Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2004). CrossRefzbMATHGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica (IIT) 2019

Authors and Affiliations

  1. 1.Univ Rennes, CNRS, IRMAR - UMR 6625RennesFrance

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