Summary.
Consider a reaction-diffusion system of the form u t −MΔu+F(u)=0, where M is a m×m matrix whose spectrum is included in {ℛz>0}. We approximate it by the Peaceman-Rachford approximation defined by P(t)=(1+tF/2)−1(1+tM Δ;/2)(1−tM Δ/2)−1(1−tF/2). We prove convergence of this scheme and show that it is of order two.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akrivis, G., Crouzeix, M., Makridakis, C.: Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comp. 67(222), 457–477 (1998)
Bergh, J., Löfström,J.: Interpolation spaces. An introduction. Springer-Verlag, Berlin 1976, Grundlehren der Mathematischen Wissenschaften, No. 223
Besse, C., Bidégaray, B., Descombes, S.: Order estimates in time of splitting methods for the nonlinear Schrödinger equation. English SIAM J. Numer. Anal. 40(1), 26–40 (2002)
Chiu, C., Walkington, N.: An ADI method for hysteretic reaction-diffusion systems. SIAM J. Numer. Anal. 34(3), 1185–1206 (1997)
de Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347(5), 1533–1589 (1995)
Descombes, S.: Convergence of a splitting method of high order for reaction-diffusion systems. Math. Comp. 70(236) 1481–1501 (2001) (electronic)
Descombes, S., Schatzman, M.: Strang's formula for holomorphic semi-groups. J. Math. Pures Appl. 81(1) 93–114 (2002)
Dia, B.O.: Méthodes de directions alternées d'ordre élevé en temps. PhD thesis, Université Claude-Bernard Lyon 1, France, June 1996
Dia, B.O., Schatzman, M.: An estimate of the Kac transfer operator. J. Funct. Anal. 145(1), 108–135 (1997)
Douglas Jr., J.: On the numerical integration of ∂2 u/∂ x 2+∂2 u/∂ y 2=∂ u/∂ t by implicit methods. J. Soc. Indust. Appl. Math. 3, 42–65 (1955)
Henry, D.: Geometric theory of semilinear parabolic equations. Berlin: Springer-Verlag, 1981
Kato, T.: Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132. New York: Springer-Verlag Inc. New York, 1966
Murray, J.D.: Mathematical biology. Berlin: Springer-Verlag, second edition, 1993
Peaceman, D.W., Rachford Jr., H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appl. Math. 3, 28–41 (1955)
Schatzman, M.: Stability of the Peaceman-Rachford formula. J. Funct. Anal. 162, 219–255 (1999)
Strang, G.: Accurate partial difference methods. I. Linear Cauchy problems. Arch. Rational Mech. Anal. 12, 392–402 (1963)
Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)
Thual, O., Fauve, S.: Localized structures generated by subcritical instabilities. J. Phys France. 49, 1829–1833 (1988)
Author information
Authors and Affiliations
Corresponding author
Additional information
We would like to thank Prof. Michelle Schatzman for reading carefully the manuscript.
Mathematics Subject Classification (2000): 65M12
Rights and permissions
About this article
Cite this article
Descombes, S., Ribot, M. Convergence of the Peaceman-Rachford approximation for reaction-diffusion systems. Numer. Math. 95, 503–525 (2003). https://doi.org/10.1007/s00211-002-0434-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-002-0434-9