An efficient numerical scheme for precise time integration of a diffusion-dissolution/precipitation chemical system

Faugeras, Blaise; Pousin, Jérôme; Fontvieille, Franck

doi:10.1090/S0025-5718-05-01782-5

An efficient numerical scheme for precise time integration of a diffusion-dissolution/precipitation chemical system
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by Blaise Faugeras, Jérôme Pousin and Franck Fontvieille;

Math. Comp. 75 (2006), 209-222

DOI: https://doi.org/10.1090/S0025-5718-05-01782-5

Published electronically: September 29, 2005

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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.

References

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. MR 1921908, DOI 10.1137/S0036142900381497
Stéphane Descombes, Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comp. 70 (2001), no. 236, 1481–1501. MR 1836914, DOI 10.1090/S0025-5718-00-01277-1
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.
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
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, DOI 10.1007/978-3-662-09947-6
Tobias Jahnke and Christian Lubich, Error bounds for exponential operator splittings, BIT 40 (2000), no. 4, 735–744. MR 1799313, DOI 10.1023/A:1022396519656
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.
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, DOI 10.1016/S0377-0427(97)00049-6
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
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, DOI 10.1023/A:1015140328635
Lawrence F. Shampine, Interpolation for Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), no. 5, 1014–1027. MR 799125, DOI 10.1137/0722060
Bruno Sportisse, An analysis of operator splitting techniques in the stiff case, J. Comput. Phys. 161 (2000), no. 1, 140–168. MR 1762076, DOI 10.1006/jcph.2000.6495
Gilbert Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506–517. MR 235754, DOI 10.1137/0705041