Abstract
We construct and analyse a fully-explicit finite-difference scheme for a two-dimensional parabolic equation with nonlocal integral conditions. The main attention is paid to the stability of the scheme. We apply the stability analysis technique which is based on the investigation of the spectral structure of the transition matrix of a finite-difference scheme and demonstrate that depending on the parameters of nonlocal conditions the proposed method can be stable or unstable. The results of numerical experiment with one test problem are also presented and they validate theoretical results.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
References
Atkinson, K.E.: An Introduction to Numerical Analysis. John Willey & Sons, New York (1978)
Cahlon, B., Kulkarni, D.M., Shi, P.: Stepwise stability for the heat equation with a nonlocal constraint. SIAM J. Numer. Anal. 32(2), 571–593 (1995)
Dehghan, M.: Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition. Math. Comput. Simulat. 49(4–5), 331–349 (1999)
Jesevičiūtė, Ž., Sapagovas, M.: On the stability of finite-difference schemes for parabolic equations subject to integral conditions with applications to thermoelasticity. Comput. Methods Appl. Math. 8(4), 360–373 (2008)
Sajavičius, S.: On the stability of alternating direction method for two-dimensional parabolic equation with nonlocal integral conditions. In: Kleiza, V., Rutkauskas, S., Štikonas, A. (eds.) Proceedings of International Conference Differential Equations and their Applications (DETA 2009), pp. 87–90. Panevėžys, Lithuania (2009)
Sajavičius, S.: The stability of finite-difference scheme for two-dimensional parabolic equation with nonlocal integral conditions. In: Damkilde, L., Andersen, L., Kristensen, A.S., Lund, E. (eds.) Proceedings of the Twenty Second Nordic Seminar on Computational Mechanics / DCE Technical Memorandum, vol. 11, pp. 87–90. Aalborg, Denmark (2009)
Sajavičius, S.: On the stability of locally one-dimensional method for two-dimensional parabolic equation with nonlocal integral conditions. In: Pereira, J.C.F., Sequeira, A., Pereira, J.M.C. (eds.) Proceedings of the V European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2010). CD-ROM, Paper ID 01668, 11 p. Lisbon, Portugal (2010)
Sajavičius, S.: On the eigenvalue problems for differential operators with coupled boundary conditions. Nonlinear Anal., Model. Control 15(4), 493–500 (2010)
Sajavičius, S.: On the eigenvalue problems for finite-difference operators with coupled boundary conditions. Šiauliai Math. Semin. 5(13), 87–100 (2010)
Sajavičius, S., Sapagovas, M.: Numerical analysis of the eigenvalue problem for one-dimensional differential operator with nonlocal integral conditions. Nonlinear Anal., Model. Control 14(1), 115–122 (2009)
Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc., New York–Basel (2001)
Sapagovas, M.: On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems. Lith. Math. J. 48(3), 339–356 (2008)
Sapagovas, M.P.: On stability of finite-difference schemes for one-dimensional parabolic equations subject to integral condition. Obchysl. Prykl. Mat. 92, 77–90 (2005)
Sapagovas, M., Kairytė, G., Štikonienė, O., Štikonas, A.: Alternating direction method for a two-dimensional parabolic equation with a nonlocal boundary condition. Math. Model. Anal. 12(1), 131–142 (2007)
Yang, W.Y., Cao, W., Chung, T.-S., Morris, J.: Applied Numerical Methods Using MATLAB®. John Willey & Sons, New York (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sajavičius, S. (2011). On the Stability of Fully-Explicit Finite-Difference Scheme for Two-Dimensional Parabolic Equation with Nonlocal Conditions. In: Murgante, B., Gervasi, O., Iglesias, A., Taniar, D., Apduhan, B.O. (eds) Computational Science and Its Applications - ICCSA 2011. ICCSA 2011. Lecture Notes in Computer Science, vol 6785. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21898-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-21898-9_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21897-2
Online ISBN: 978-3-642-21898-9
eBook Packages: Computer ScienceComputer Science (R0)