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An efficient numerical scheme for precise time integration of a diffusion-dissolution/precipitation chemical system
Authors:
Blaise Faugeras, Jérôme Pousin and Franck Fontvieille
Journal:
Math. Comp. 75 (2006), 209-222
MSC (2000):
Primary 65M12, 65G99, 35K57
Posted:
September 29, 2005
MathSciNet review:
2176396
Full-text PDF Free Access
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Additional Information
Abstract: A numerical scheme based on an operator splitting method and a dense output event location algorithm is proposed to integrate a diffusion-dissolution/precipitation chemical initial-boundary value problem with jumping nonlinearities. The numerical analysis of the scheme is carried out and it is proved to be of order 2 in time. This global order estimate is illustrated numerically on a test case.
- 1.
Christophe
Besse, Brigitte
Bidégaray, and Stéphane
Descombes, Order estimates in time of splitting methods for the
nonlinear Schrödinger equation, SIAM J. Numer. Anal.
40 (2002), no. 1, 26–40 (electronic). MR 1921908
(2003k:65099), http://dx.doi.org/10.1137/S0036142900381497
- 2.
Stéphane
Descombes, Convergence of a splitting method of
high order for reaction-diffusion systems, Math. Comp. 70 (2001), no. 236, 1481–1501 (electronic). MR 1836914
(2002c:65152), http://dx.doi.org/10.1090/S0025-5718-00-01277-1
- 3.
S. Descombes and M. Massot, Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction, Numerische Mathematik (2004) Vol. 97, No. 4, 667-698.
- 4.
E.
Hairer, S.
P. Nørsett, and G.
Wanner, Solving ordinary differential equations. I, 2nd ed.,
Springer Series in Computational Mathematics, vol. 8, Springer-Verlag,
Berlin, 1993. Nonstiff problems. MR 1227985
(94c:65005)
- 5.
E.
Hairer and G.
Wanner, Solving ordinary differential equations. II, Springer
Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin,
1991. Stiff and differential-algebraic problems. MR 1111480
(92a:65016)
- 6.
Tobias
Jahnke and Christian
Lubich, Error bounds for exponential operator splittings, BIT
40 (2000), no. 4, 735–744. MR 1799313
(2001k:65143), http://dx.doi.org/10.1023/A:1022396519656
- 7.
E. Maisse, Analyse et simulations numériques de phénomènes de diffusion-dissolution/précipitation en milieux poreux, appliquées au stockage de déchets, Ph.D. thesis, Université Claude Bernard Lyon I, 1998.
- 8.
E.
Maisse and J.
Pousin, Diffusion and dissolution/precipitation in an open porous
reactive medium, J. Comput. Appl. Math. 82 (1997),
no. 1-2, 279–290. 7th ICCAM 96 Congress (Leuven). MR 1473546
(98g:35170), http://dx.doi.org/10.1016/S0377-0427(97)00049-6
- 9.
G.
I. Marchuk, Splitting and alternating direction methods,
Handbook of numerical analysis, Vol. I, Handb. Numer. Anal., I,
North-Holland, Amsterdam, 1990, pp. 197–462. MR
1039325
- 10.
Michelle
Schatzman, Toward non commutative numerical analysis: high order
integration in time, Proceedings of the Fifth International Conference
on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), 2002,
pp. 99–116. MR
1910554, http://dx.doi.org/10.1023/A:1015140328635
- 11.
Lawrence
F. Shampine, Interpolation for Runge-Kutta methods, SIAM J.
Numer. Anal. 22 (1985), no. 5, 1014–1027. MR 799125
(86j:65014), http://dx.doi.org/10.1137/0722060
- 12.
Bruno
Sportisse, An analysis of operator splitting techniques in the
stiff case, J. Comput. Phys. 161 (2000), no. 1,
140–168. MR 1762076
(2001f:65107), http://dx.doi.org/10.1006/jcph.2000.6495
- 13.
Gilbert
Strang, On the construction and comparison of difference
schemes, SIAM J. Numer. Anal. 5 (1968),
506–517. MR 0235754
(38 #4057)
- 1.
- C. Besse, B. Bidegaray, and S. Descombes, Order estimates in time of splitting methods for the nonlinear schrödinger equation, SIAM J. Numer. Anal. 40 (2002), no. 5, 26-40. MR 1921908 (2003k:65099)
- 2.
- S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comp. 70 (2001), no. 236, 1484-1501.MR 1836914 (2002c:65152)
- 3.
- S. Descombes and M. Massot, Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction, Numerische Mathematik (2004) Vol. 97, No. 4, 667-698.
- 4.
- E. Hairer, S.P. Norsett, and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer Series in Computational Mathematics, Springer Verlag, 1993. MR 1227985 (94c:65005)
- 5.
- E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, Springer Verlag, 1993. MR 1111480 (92a:65016)
- 6.
- C. Lubich and T. Jahnke, Error bounds for exponential operator splitting, Technical report, Universitat Tubingen, Germany, 1999; BIT 40 (2000), 735-744.MR 1799313 (2001k:65143)
- 7.
- E. Maisse, Analyse et simulations numériques de phénomènes de diffusion-dissolution/précipitation en milieux poreux, appliquées au stockage de déchets, Ph.D. thesis, Université Claude Bernard Lyon I, 1998.
- 8.
- E. Maisse and J. Pousin, Diffusion and dissolution/precipitation in an open porous reactive medium, J. Comp. Appl. Math. 82 (1997), 279-280.MR 1473546 (98g:35170)
- 9.
- G.I. Marchuk, Splitting and alternating direction methods, Handbook of numerical analysis, vol. I, North-Holland, Amsterdam, 1990, pp. 197-462. MR 1039325
- 10.
- M. Schatzman, Toward non commutative numerical analysis : high order integration in time, Journal of Scientific Computing 17 (2002), no. 1-3, 107-125. MR 1910554
- 11.
- L.F. Shampine, Interpolation for Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), 1014-1027. MR 0799125 (86j:65014)
- 12.
- B. Sportisse, An analysis of operator splitting techniques in the stiff case, J. Comput. Phys. 161 (2000), no. 1, 140-168. MR 1762076 (2001f:65107)
- 13.
- G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506-517. MR 0235754 (38:4057)
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Additional Information
Blaise Faugeras
Affiliation:
CNRS I35, Les Algorithmes, 2000 Route des Lucioles, BP 121, 06903 Sophia Antipolis cedex France
Email:
Blaise.Faugeras@unice.fr
Jérôme Pousin
Affiliation:
MAPLY, Centre de Mathématique INSA de Lyon, Bat. Léonard de Vinci, 21, Av. Jean Capelle, 69100 Villeurbanne Cedex, France
Email:
Jerome.Pousin@insa-lyon.fr
Franck Fontvieille
Affiliation:
MAPLY, Centre de Mathématique INSA de Lyon, Bat. Léonard de Vinci, 21, Av. Jean Capelle, 69100 Villeurbanne Cedex, France
Email:
Franck.Fontvieille@insa-lyon.fr
DOI:
http://dx.doi.org/10.1090/S0025-5718-05-01782-5
PII:
S 0025-5718(05)01782-5
Keywords:
Numerical time integration,
operator splitting,
dense output,
high order,
error analysis,
reaction-diffusion,
jumping nonlinearities
Received by editor(s):
December 2, 2003
Received by editor(s) in revised form:
September 20, 2004
Posted:
September 29, 2005
Article copyright:
© Copyright 2005 American Mathematical Society
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