Abstract
In the present paper the initial-boundary value problem for multidimensional hyperbolic equation with Dirichlet condition is considered. The third and fourth orders of accuracy difference schemes for the approximate solution of this problem are presented and the stability estimates for the solutions of these difference schemes are obtained. Some results of numerical experiments are presented in order to support theoretical statements.
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
Lamb, H.: Hydrodynamics. Cambridge University Press, Cambridge (1993)
Lighthill, J.: Waves in Fluids. Cambridge University Press, Cambridge (1978)
Hudson, J.A.: The Excitation and Propagation of Elastic Waves. Cambridge University Press, Cambridge (1980)
Jones, D.S.: Acoustic and Electromagnetic Waves. The Clarendon Press/Oxford University Press, New York (1986)
Taflove, A.: Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House, Boston (1995)
Ciment, M., Leventhal, S.H.: A note on the operator compact implicit method for the wave equation. Mathematics of Computation 32(141), 143–147 (1978)
Twizell, E.H.: An explicit difference method for the wave equation with extended stability range. BIT 19(3), 378–383 (1979)
Ashyralyev, A., Sobolevskii, P.E.: Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations. Discrete Dynamics Nature and Society 2(1), 183–213 (2005)
Ashyralyev, A., Sobolevskii, P.E.: New Difference Schemes for Partial Differential Equations, Operator Theory: Advances and Applications. Birkhäuser, Basel (2004)
Ashyralyev, A., Sobolevskii, P.E.: A note on the difference schemes for hyperbolic equations. Abstract and Applied Analysis 6(2), 63–70 (2001)
Ashyralyev, A., Koksal, M.E., Agarwal, R.P.: A difference scheme for Cauchy problem for the hyperbolic equation with self-adjoint operator. Mathematical and Computer Modelling 52(1-2), 409–424 (2010)
Ashyralyev, A., Aggez, N.: A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations. Numerical Functional Analysis and Optimization 25(5-6), 1–24 (2004)
Ashyralyev, A., Yildirim, O.: Second order of accuracy stable difference schemes for hyperbolic problems subject to nonlocal conditions with self-adjoint operator. In: 9th International Conference of Numerical Analysis and Applied Mathematics, pp. 597–600. AIP Conference Proceedings, New York (2011)
Ashyralyev, A., Yildirim, O.: On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations. Taiwanese Journal of Mathematics 14(1), 165–194 (2010)
Sobolevskii, P.E., Chebotaryeva, L.M.: Approximate solution by method of lines of the Cauchy problem for an abstract hyperbolic equations. Izv. Vyssh. Uchebn. Zav. Matematika 5(1), 103–116 (1977)
Samarskii, A.A., Lazarov, R.D., Makarov, V.L.: Difference Schemes for Differential Equations with Generalized Solutions. Vysshaya Shkola, Moscow (1987) (in Russian)
Ashyralyev, A., Ozdemir, Y.: On stable implicit difference scheme for hyperbolic-parabolic equations in a Hilbert space. Numerical Methods for Partial Differential Equations 25(1), 1110–1118 (2009)
Ashyralyev, A., Gercek, O.: On second order of accuracy difference scheme of the approximate solution of nonlocal elliptic-parabolic problems. Abstract and Applied Analysis 2010 (ID 705172), 17 p. (2010)
Ashyralyev, A., Judakova, G., Sobolevskii, P.E.: A note on the difference schemes for hyperbolic-elliptic equations. Abstract and Applied Analysis 2006, 01–13 (2006)
Sobolevskii, P.E.: Difference Methods for the Approximate Solution of Differential Equations. Izdat. Gosud. Univ: Voronezh (1975)
Gordezani, D., Meladze, H., Avalishvili, G.: On one class of nonlocal in time problems for first-order evolution equations. Zh. Obchysl. Prykl. Mat. 88(1), 66–78 (2003)
Sapagovas, M.: On stability of the finite difference schemes for a parabolic equations with nonlocal condition. Journal of Computer Applied Mathematics 1(1), 89–98 (2003)
Mitchell, A.R., Griffiths, D.F.: The Finite Difference Methods in Partial Differential Equations. Wiley, New York (1980)
Lax, P.D., Wendroff, B.: Difference schemes for hyperbolic equations with high order of accuracy. Comm. Pure Appl. Math. 17, 381–398 (1964)
Gustafson, B., Kreiss, H.O., Oliger, J.: Time dependent problems and difference methods. Wiley, New York (1995)
MartÃn-Vaquero, J., Queiruga-Dios, A., Encinas, A.H.: Numerical algorithms for diffusion-reaction problems with non-classical conditions. Applied Mathematics and Computation 218(9), 5487–5495 (2012)
Simos, T.E.: A finite-difference method for the numerical solution of the Schrodinger equation. Journal of Computational and Applied Mathematics 79(4), 189–205 (1997)
Fattorini, H.O.: Second Order Linear Differential Equations in Banach Space. Notas de Matematica, North-Holland (1985)
Krein, S.G.: Linear Differential Equations in a Banach Space. Nauka, Moscow (1966)
Piskarev, S., Shaw, Y.: On certain operator families related to cosine operator function. Taiwanese Journal of Mathematics 1(4), 3585–3592 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ashyralyev, A., Yildirim, O. (2013). High Order Accurate Difference Schemes for Hyperbolic IBVP. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2012. Lecture Notes in Computer Science, vol 8236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41515-9_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-41515-9_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41514-2
Online ISBN: 978-3-642-41515-9
eBook Packages: Computer ScienceComputer Science (R0)