Solving the systems of equations arising in the discretization of some nonlinear p.d.e.'s by implicit Runge-Kutta methods

Georgios Akrivis; Vassilios A. Dougalis; Ohannes Karakashian

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (1997)

  • Volume: 31, Issue: 2, page 251-287
  • ISSN: 0764-583X

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Akrivis, Georgios, Dougalis, Vassilios A., and Karakashian, Ohannes. "Solving the systems of equations arising in the discretization of some nonlinear p.d.e.'s by implicit Runge-Kutta methods." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 31.2 (1997): 251-287. <http://eudml.org/doc/193837>.

@article{Akrivis1997,
author = {Akrivis, Georgios, Dougalis, Vassilios A., Karakashian, Ohannes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {implicit Runge-Kutta method; nonlinear Schrödinger equation; iterative methods; nonlinear equations; nonlinear evoluton equations; Newton method; Korteweg-de Vries equation},
language = {eng},
number = {2},
pages = {251-287},
publisher = {Dunod},
title = {Solving the systems of equations arising in the discretization of some nonlinear p.d.e.'s by implicit Runge-Kutta methods},
url = {http://eudml.org/doc/193837},
volume = {31},
year = {1997},
}

TY - JOUR
AU - Akrivis, Georgios
AU - Dougalis, Vassilios A.
AU - Karakashian, Ohannes
TI - Solving the systems of equations arising in the discretization of some nonlinear p.d.e.'s by implicit Runge-Kutta methods
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 1997
PB - Dunod
VL - 31
IS - 2
SP - 251
EP - 287
LA - eng
KW - implicit Runge-Kutta method; nonlinear Schrödinger equation; iterative methods; nonlinear equations; nonlinear evoluton equations; Newton method; Korteweg-de Vries equation
UR - http://eudml.org/doc/193837
ER -

References

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  1. [1] G. D. AKRIVIS, V. A. DOUGALIS and O. A. KARAKASHIAN, 1991, On fully discrete Galerkin methods of second-order temporal accuracy for the Nonlinear Schrödinger Equation, Numer. Math., 59, pp. 31-53. Zbl0739.65096MR1103752
  2. [2] R. ALEXANDER, 1991, The modified Newton method in the solution of stiff ordinary differential equations, Math. Comp., 57, pp. 673-701. Zbl0734.65060MR1094939
  3. [3] G. A. BAKER, V. A. DOUGALIS and O. A. KARAKASHIAN, 1983, Convergence of Galerkin approximations for the Korteweg-de Vries equation, Math. Comp., 40, pp. 419-433. Zbl0519.65073MR689464
  4. [4] J. L. BONA, V. A. DOUGALIS, O. A. KARAKASHIAN and W. R. McKlNNEY, 1995, Conservative high order numerical methods for the generalized Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London Ser. A, 351, pp. 107-164. Zbl0824.65095MR1336983
  5. [5] J. L. BONA and R. SMITH, 1975, The initial value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 298, pp. 555-604. Zbl0306.35027MR385355
  6. [6] J. C. BUTCHER, 1987, The numerical analysis of ordinary differerential equations. Runge-Kutta and general linear methods, John Wiley & Sons. Zbl0616.65072MR878564
  7. [7] M. CROUZEIX, W. H. HUNDSDORFER and M. N. SPIJKER, 1983, On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods, BIT, 23, pp. 84-91. Zbl0506.65030MR689606
  8. [8] V. A. DOUGALIS and O. A. KARAKASHIAN, 1985, On some high order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation, Math. Comp., 45, pp. 329-345. Zbl0609.65064MR804927
  9. [9] E. HAIRER and G. WANNER, 1991, Solving ordinary differential equations II. Stiff and differential-algebraic problems, Springer series in Computational Mathematics, Springer Verlag. Zbl0729.65051MR1111480
  10. [10] O. KARAKASHIAN and W. RUST, 1988, On the parallel implementation of implicit Runge-Kuta methods, SIAM J. Sci. Sta. Comput., 9, pp. 1085-1090. Zbl0664.65068MR963856
  11. [11] O. KARAKASHIAN, G. D. AKRIVIS and V. A. DOUGALIS, 1993, On optimal-order error estimates for the Nonlinear Schrödinger Equation, SIAM J. Numer. Anal., 30, pp. 377-400. Zbl0774.65091MR1211396
  12. [12] O. KARAKASHIAN and W. MCKINNEY, 1990, On optimal high order in time approximations for the Korteweg-de Vries equation, Math. Comp., 55, pp. 473-496. Zbl0725.65107MR1035935
  13. [13] J. M. SANZ-SERNA and D. F. GRIFFITHS, 1991, A new class of results for the algebraic equations of implicit Runge-Kutta processes, IMA Journal of Numerical Analysis, 11, pp. 449-455. Zbl0738.65067MR1135198
  14. [14] V. THOMÉE and B. WENDROFF, 1974, Convergence estimates for Galerkin methods for variable coefficient initial value problems, SIAM J. Numer. Anal., 11, pp. 1059-1068. Zbl0292.65052MR371088

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