Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T16:55:34.216Z Has data issue: false hasContentIssue false
  • English
  • Français

A Second-Order Three-Level Difference Scheme for a Magneto-Thermo-Elasticity Model

Published online by Cambridge University Press:  21 May 2014

Hai-Yan Cao ,
Zhi-Zhong Sun  and
Xuan Zhao
Hai-Yan Cao*
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
Zhi-Zhong Sun*
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
Xuan Zhao*
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
*
Email: maths@seu.edu.cn
Corresponding author. Email: zzsun@seu.edu.cn
Email: xuanzhao11@gmail.com or xzhao@seu.edu.cn
Get access

Abstract

This article deals with the numerical solution to the magneto-thermo-elasticity model, which is a system of the third order partial differential equations. By introducing a new function, the model is transformed into a system of the second order generalized hyperbolic equations. A priori estimate with the conservation for the problem is established. Then a three-level finite difference scheme is derived. The unique solvability, unconditional stability and second-order convergence in L∞-norm of the difference scheme are proved. One numerical example is presented to demonstrate the accuracy and efficiency of the proposed method.

Keywords

65M0665M1265M1278M2080M20Magneto-thermo-elasticityconservationfinite differencesolvabilitystabilityconvergence
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 3 , June 2014 , pp. 281 - 298
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Chandrasekharaiah, D. S., Thermoelasticity with second sound: a rewiew, Appl. Mech. Rev., 39(3) (1986), pp. 355–376.Google Scholar
[2]Chandrasekharaiah, D. S., Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev., 51(12) (1998), pp. 705–729.Google Scholar
[3]Hetnarski, R. B. and Ignaczak, J., Generalized thermoelasticity, J. Therm. Stresses, 22 (1999), pp. 451–476.Google Scholar
[4]Ootao, Y., Ishihara, M. and Noda, K., Transient thermal stress analysis of a functionally graded magneto-electro-thermoelastic strip due to nonuniform surface heating, Theor. Appl. Fract. Mec., 55(3) (2011), pp. 206–212.Google Scholar
[5]Messaoudi, S. and Al-Shehri, A., General boundary stabilization of memory-type thermoelasticity, J. Math. Phys., 51(10) (2010), 103514.Google Scholar
[6]Charao, R., Oliveira, J. and Menzala, G., Energy decay rates of magnetoelastic waves in a bounded conductive medium, Discrete Contin. Dyn. Syst., 25(3) (2009), pp. 797–821.Google Scholar
[7]Wang, X. and Zhou, Y., A generalized variational model of magneto-thermo-elasticityfor nonlinearly magnetized ferroelastic bodies, Int. J. Eng. Sci., 40(17) (2002), pp. 1957–1973.Google Scholar
[8]Rivera, J. Munoz and Racke, R., Magneto-thermo-elasticity-large-time behavior for linear sys-tems, Adv. Differential Equations, 6(3) (2001), pp. 359–384.Google Scholar
[9]Hinojosa, J. H. and Mickus, K. L., Thermoelastic modeling of lithospheric uplift: a finite- difference numerical solution, Comput. Geosci., 28(2) (2002), pp. 155–167.CrossRefGoogle Scholar
[10]Abd-Alla, A. M., Salama, A. A., Abd-Ei-Salam, M. R. and Hosham, H. A., An implicit finite-difference method for solving the transient coupled thermoelasticity of an annular fin, Appl. Math. Inf. Sci., 1(1) (2007), pp. 79–93.Google Scholar
[11]Jeseviciūte, Z. and Sapagovas, M., the stability of finite-difference schemes for parabolic equations subject to integral conditions with applications to thermoelasticity, Comput. Methods Appl. Math., 8(4) (2008), pp. 360–373.Google Scholar
[12]Abd-Ei-Salam, M. R., Abd-Alla, A. M. and Hosham, H. A., A numerical solution of magneto-thermoelastic problem in non-homogeneous isotropic cylinder by the finite-difference method, Appl. Math. Model., 31 (2007), pp. 1662–1670.Google Scholar
[13]Rincon, M. A., Santos, B. S. and Imaco, J., Numerical method, existence and uniqueness for thermoelasticity system with moving boundary, Comput. Appl. Math., 24(3) (2005), pp. 439–460.Google Scholar
[14]Zhao, Y. Q. and Lin, W., Numerical solution to problems concerning quasi-static thermoelasticity, Appl. Anal., 57(1-2) (1995), pp. 221–234.Google Scholar
[15]Copetti, M. I. M. and French, D. A., Numerical approximation and error control for a thermoelastic contact problem, Appl. Numer. Math., 55(4) (2005), pp. 439–457.Google Scholar
[16]Copetti, M. I. M., Numerical analysis of a unilateral problem in planar thermoelasticity, SIAM J. Numer. Anal., 45(6) (2007), pp. 2637–2652.Google Scholar
[17]Yu, G. Ispolov, Postoyalkina, E. A. and Shabrov, N. N., New methods for the numerical integration of equations of the coupled thermoelasticity problem, Differ. Uravn. Protsessy Upr., 3 (1998), pp. 1–17.Google Scholar
[18]Kasti, N. A., Lubliner, J. and Taylor, R. L., A numerical-solution for the coupled quasi-static linear thermoelastic problem, J. Therm. Stresses, 14(3) (1991), pp. 333–349.Google Scholar
[19]Sumi, N., Numerical solutions of thermoelastic wave problems by the method of characteristics, J. Therm. Stresses, 24(6) (2001), pp. 509–530.Google Scholar
[20]Das, P. and Kanoria, M., Magneto-thermo-elastic waves in an infinite perfectly conducting elastic solid with energy dissipation, Appl. Math. Mech. Eng. Ed., 30(2) (2009), pp. 221–228.Google Scholar
[21]Green, A. E. and Naghdi, P. M., A re-examination of the basic postulate of thermo-mechanics, Proc. Royal Soc. London, 432 (1991), pp. 171–194.Google Scholar
[22]Green, A. E. and Naghdi, P. M., An unbounded heat wave in an elastic solid, J. Therm. Stresses, 15 (1992), pp. 253–264.Google Scholar
[23]Green, A. E. and Naghdi, P. M., Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), pp. 189–208.Google Scholar
[24]Sun, Z. Z., Numerical solution to Partial Differential Equations, 2nd edition, Science Press, Beijing, 2012.Google Scholar