We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem for hyperbolic heat transfer equation. New stability and approximation error estimates are proved and it is noted that some statements given in the above papers should be modified and improved. Finally, two robust finite difference schemes are proposed, that can be used for both, the hyperbolic and parabolic heat transfer equations. Results of numerical experiments are presented.
@article{bwmeta1.element.doi-10_2478_s11533-011-0056-5,
author = {Raimondas \v Ciegis and Aleksas Mirinavi\v cius},
title = {On some finite difference schemes for solution of hyperbolic heat conduction problems},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {1164-1170},
zbl = {1236.65109},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0056-5}
}
Raimondas Čiegis; Aleksas Mirinavičius. On some finite difference schemes for solution of hyperbolic heat conduction problems. Open Mathematics, Tome 9 (2011) pp. 1164-1170. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0056-5/
[1] Čiegis R., Numerical solution of hyperbolic heat conduction equation, Math. Model. Anal., 2009, 14(1), 11–24 http://dx.doi.org/10.3846/1392-6292.2009.14.11-24 | Zbl 1175.80020
[2] Čiegis R., Dement’ev A., Jankevičiūtė G., Numerical analysis of the hyperbolic two-temperature model, Lith. Math. J., 2008, 48(1), 46–60 http://dx.doi.org/10.1007/s10986-008-0005-6 | Zbl 1211.65107
[3] Kalis H., Buikis A., Method of lines and finite difference schemes with the exact spectrum for solution the hyperbolic heat conduction equation, Math. Model. Anal., 2011, 16(2), 220–232 http://dx.doi.org/10.3846/13926292.2011.578677 | Zbl 1220.65114
[4] Manzari Meh.T., Manzari Maj.T., On numerical solution of hyperbolic heat conduction, Comm. Numer. Methods Engrg., 1999, 15(12), 853–866 http://dx.doi.org/10.1002/(SICI)1099-0887(199912)15:12<853::AID-CNM293>3.0.CO;2-V
[5] Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649 http://dx.doi.org/10.1002/num.20003 | Zbl 1062.65086
[6] Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430 http://dx.doi.org/10.1016/j.apnum.2008.09.001 | Zbl 1162.65398
[7] Samarskii A.A., The Theory of Difference Schemes, Monogr. Textbooks Pure Appl. Math., 240, Marcel Dekker, New York, 2001 http://dx.doi.org/10.1201/9780203908518
[8] Samarskii A.A., Matus P.P., Vabishchevich P.N., Difference Schemes with Operator Factors, Math. Appl., 546, Kluwer, Dordrecht, 2002 | Zbl 1018.65103
[9] Sarra S.A., Spectral methods with postprocessing for numerical hyperbolic heat transfer, Numerical Heat Transfer, Part A, 2003, 43(7), 717–730 http://dx.doi.org/10.1080/713838126
[10] Shen W., Han S., A numerical solution of two-dimensional hyperbolic heat conduction with non-linear boundary conditions, Heat and Mass Transfer, 2003, 39(5–6), 499–507