Convergence of the Peaceman-Rachford approximation for reaction-diffusion systems

Summary.Consider a reaction-diffusion system of the form ut−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+tMΔ;/2)(1−tMΔ/2)−1(1−tF/2). We prove convergence of this scheme and show that it is of order two.

[1]  Michelle Schatzman,et al.  An Estimate of the Kac Transfer Operator , 1997 .

[2]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[3]  Christophe Besse,et al.  Order Estimates in Time of Splitting Methods for the Nonlinear Schrödinger Equation , 2002, SIAM J. Numer. Anal..

[4]  P. H. Müller,et al.  T. Kato, Perturbation theory for linear operators. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 132) XX + 592 S. m. 3 Fig. Berlin/Heidelberg/New York Springer-Verlag. Preis geb. DM 79,20 , 1967 .

[5]  Charalambos Makridakis,et al.  Implicit-explicit multistep finite element methods for nonlinear parabolic problems , 1998, Math. Comput..

[6]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[7]  Michelle Schatzman,et al.  Geometrical evolution of developed interfaces , 1995, Emerging applications in free boundary problems.

[8]  Noel J. Walkington,et al.  An ADI Method for Hysteretic Reaction-Diffusion Systems , 1997 .

[9]  Stéphane Descombes,et al.  Strang's formula for holomorphic semi-groups , 2002 .

[10]  G. Strang.  Accurate partial difference methods I: Linear cauchy problems , 1963 .

[11]  Michelle Schatzman,et al.  Stability of the Peaceman–Rachford Approximation , 1999 .

[12]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[13]  Stéphane Descombes,et al.  Convergence of a splitting method of high order for reaction-diffusion systems , 2001, Math. Comput..

[14]  Boun Oumar Dia.  Méthodes de directions alternées d'ordre élevé en temps , 1996 .

[15]  S. Fauve,et al.  Localized structures generated by subcritical instabilities , 1988 .

[16]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[17]  Gilbert Strang,et al.  Accurate partial difference methods , 1964 .