BAIL 2006 Book of Abstracts - Institut für Numerische und ...

C. CLAVERO, J.L. GRACIA, F. LISBONA: A second order uniform convergent
method for a singularly perturbed parabolic system of reaction-diffusion type
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In [7] and [6] it was developed a first order uniformly convergent method in cases [ii.] and [iii.]
respectively. In [8], [4] and [5] a second order uniformly convergent scheme was obtained for
cases [i.], [ii.] and [iii.] respectively.
In [3] a decomposition of the exact solution of problem (1), into its regular and singular
components, was given. From this decomposition it follows which is the asymptotic behaviour
of each one of these components with respect to the singular perturbation parameters ε1 and ε2.
Moreover, in that work a first order in time and second order in space (except by a logarithmic
factor) uniformly convergent method was developed, using the classical Euler and central differences
discretizations respectively. In order to increase the order of uniform convergence of this
numerical scheme, here we replace the Euler scheme by the Crank-Nicolson method in the time
discretization. This method has been used in the framework of singularly perturbed problem; for
instance, in [2] to solve 1D evolutionary problems of convection–diffusion type. Some numerical
experiments will be showed, which illustrate in practice the improvement in the uniform order
of convergence of the new scheme.
Keywords: Singular perturbation, reaction-diffusion problems, uniform convergence, coupled
system, Shishkin mesh.
AMS classification: 65N12, 65N30, 65N06
References
[1] G.I. Barenblatt, I.P. Zheltov and I.N. Kochina, “Basic concepts in the theory of seepage of
homogeneous liquids in fissured rocks”, J. Appl. Math. and Mech., 24, 1286–1303 (1960).
[2] C. Clavero, J.L. Gracia and J.C. Jorge, “Second order numerical methods for one–
dimensional parabolic singularly perturbed problems with regular layers”, Numerical Methods
for Partial Differential Equations, 21 149–169 (2005).
[3] J.L. Gracia and F. Lisbona “A uniformly convergent scheme for a system of reaction–
diffusion equations”, submitted.
[4] T. Linß and N. Madden, “An improved error estimate for a numerical method for a system
of coupled singularly perturbed reaction–diffusion equations”, Comp. Meth. Appl. Math. 3
417–423 (2003).
[5] T. Linß and N. Madden, “Accurate solution of a system of coupled singularly perturbed
reaction-diffusion equations”, Computing, 73, 121–133 (2004).
[6] N. Madden and M. Stynes, “A uniformly convergent numerical method for a coupled system
of two singularly perturbed linear reaction-diffusion problems“, IMA J. Numer. Anal., 23,
627–644 (2003).
[7] S. Matthews, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A parameter robust numerical
method for a system of singularly perturbed ordinary differential equations, in: Analytical
and Numerical Methods for Convection–Dominated and Singularly Perturbed Problems
(J.J.H. Miller, G.I. Shishkin and L. Vulkov, eds.), Nova Science Publishers, New York, 2000,
219–224.
[8] S. Matthews, E. O’Riordan and G.I. Shishkin, “A nunerical method for a system of singularly
perturbed reaction–diffusion equations“, J. Comput. Appl. Math., 145, 151–166
(2002).
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Speaker: CLAVERO, C. 78 BAIL 2006
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