# On some finite difference schemes for solution of hyperbolic heat conduction problems

Raimondas Čiegis; Aleksas Mirinavičius

Open Mathematics (2011)

- Volume: 9, Issue: 5, page 1164-1170
- ISSN: 2391-5455

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topRaimondas Čiegis, and Aleksas Mirinavičius. "On some finite difference schemes for solution of hyperbolic heat conduction problems." Open Mathematics 9.5 (2011): 1164-1170. <http://eudml.org/doc/269151>.

@article{RaimondasČiegis2011,

abstract = {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.},

author = {Raimondas Čiegis, Aleksas Mirinavičius},

journal = {Open Mathematics},

keywords = {Hyperbolic heat conduction; Finite difference method; Stability analysis; Convergence analysis; hyperbolic heat equation; finite difference method; stability analysis},

language = {eng},

number = {5},

pages = {1164-1170},

title = {On some finite difference schemes for solution of hyperbolic heat conduction problems},

url = {http://eudml.org/doc/269151},

volume = {9},

year = {2011},

}

TY - JOUR

AU - Raimondas Čiegis

AU - Aleksas Mirinavičius

TI - On some finite difference schemes for solution of hyperbolic heat conduction problems

JO - Open Mathematics

PY - 2011

VL - 9

IS - 5

SP - 1164

EP - 1170

AB - 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.

LA - eng

KW - Hyperbolic heat conduction; Finite difference method; Stability analysis; Convergence analysis; hyperbolic heat equation; finite difference method; stability analysis

UR - http://eudml.org/doc/269151

ER -

## References

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- [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 Zbl1211.65107
- [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 Zbl1220.65114
- [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 Zbl1062.65086
- [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 Zbl1162.65398
- [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 Zbl1018.65103
- [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

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