Abstract
We consider a Dirichlet problem on a ball for a singularly perturbed parabolic reaction-diffusion equation. The Laplacian in the equation is multiplied by a perturbation parameter ε 2, where ε ∈ (0,1]. The solution of such a problem exhibits the parabolic boundary layer in a neighbourhood of the ball boundary as ε→0. Using the integro-interpolational method and the condensing mesh method, we construct conservative finite difference schemes whose solutions converge ε-uniformly.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Shishkin, G.I.: Discrete Approximations of Singularly Perturbed Elliptic and Parabolic Equations. Russian Academy of Sciences, Ural Section, Ekaterinburg (1992) (in Russian)
Hemker, P.W., Shishkin, G.I., Shishkina, L.P.: ε-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems. IMA J. Numer. Anal. 20(1), 99–121 (2000)
Shishkin, G.I.: A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation. Numer. Math. Theory Methods Appl. 1(2), 214–234 (2008)
Shishkin, G.I.: Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial–boundary conditions. In: Proc. Steklov Inst. Math., vol. 2, pp. 213–230 (2007)
Grigoriev, V.A., Zorin, V.M. (eds.): Teplotechnicheskii experiment: Spravochnik. Energoizdat, Moscow (1982) (in Russian)
Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, Inc., New York (2001)
Kalitkin, N.N.: Numerical methods. Nauka, Moscow (1978) (in Russian)
Shishkin, G.I.: Approximation of solutions of singularly perturbed boundary value problems with a parabolic boundary layer. USSR Comput. Maths. Math. Phys. 29(4), 1–10 (1989)
Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems. In: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore (1996)
Il’in, A.M., Kalashnikov, A.S., Oleinik, O.A.: Second-order linear equations of parabolic type. Uspehi Mat. Nauk 17(3), 3–146 (1962) (in Russian)
Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1967)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs (1964)
Shishkin, G.I.: Grid approximation of singularly parturbed boundary value problem for quasi-linear parabolic equations in case of complete degeneracy in spatial variables. Soviet J. Numer. Anal. Math. Modelling 6(3), 243–261 (1991)
Shishkin, G.I.: Grid approximation of a singularly perturbed boundary value problem for a quasilinear elliptic equation in the case of complete degeneration. Comput. Maths. Math. Phys. 31(12), 33–46 (1991)
Samarskii, A.A.: Introduction to the Theory of Difference Schemes. Nauka, Moscow (1971) (in Russian)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shishkina, L., Shishkin, G. (2009). Grid Approximation of a Singularly Perturbed Parabolic Reaction-Diffusion Equation on a Ball. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2008. Lecture Notes in Computer Science, vol 5434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00464-3_58
Download citation
DOI: https://doi.org/10.1007/978-3-642-00464-3_58
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00463-6
Online ISBN: 978-3-642-00464-3
eBook Packages: Computer ScienceComputer Science (R0)